Nonlinear IV panel unit root testing under structural breaks in the error variance

Abstract

The paper examines the behavior of a generalized version of the nonlinear IV unit root test proposed by Chang (2002) when the series’ errors exhibit nonstationary volatility. The leading case of such nonstationary volatility concerns structural breaks in the error variance. We show that the generalized test is not robust to variance changes in general, and illustrate the extent of the resulting size distortions in finite samples. More importantly, we show that pivotality is recovered when using Eicker-White heteroskedasticity-consistent standard errors. This contrasts with the case of Dickey-Fuller unit root tests, for which Eicker-White standard errors do not produce robustness and thus require computationally costly corrections such as the (wild) bootstrap or estimation of the so-called variance profile. The pivotal versions of the generalized IV tests – with or without the correct standard errors – do however have no power in $1/T$-neighbourhoods of the null. We also study the validity of panel versions of the tests considered here.

Appendices

Appendix

Let us first analyze the behavior of the cross-products involved in the expressions of the estimators and the test statistics. To this end, some limits and definitions are provided in the following.

Lemma 1

Under Assumptions 1, 2 and 3, it holds with $\rho _{i}=1-\frac{c_{i}}{T}$ that

$$\begin{aligned} \frac{A_{i}\left(1\right)}{\overline{\omega }_{i}\sqrt{T}}\, y_{i\left[sT\right]}\Rightarrow J_{i,\eta _{i}}^{c_i}(s) \end{aligned}$$

marginally, for all $1\le i \le N$.

Proof

See Cavaliere (2004).

For $0\le \alpha <0.5,$ the asymptotic theory relies on the local time of Gaussian continuous-time processes. So let $L_{i}\left(w,s\right)$ be the chronological local time of the limiting process $J_{i,\eta _{i}}^{c_i}(s),$

$$\begin{aligned} L_{i}\left(w,s\right)=\lim _{\epsilon \rightarrow 0}\frac{1}{2\epsilon }\int _{0}^{1}\mathbb I \left(\left|J_{i,\eta _{i}}^{c_i}(s)-w\right|<\epsilon \right)\text{ d}s \end{aligned}$$

with $\mathbb I \left(\cdot \right)$ the indicator function), and denote by $L_{i}^{*}\left(w,s\right)$ the local time of the time-transformed OU process in terms of its quadratic variation,

$$\begin{aligned} L_{i}^{*}\left(w,s\right)=\lim _{\epsilon \rightarrow 0}\frac{1}{2\epsilon }\int _{0}^{1}\mathbb I \left(\left|J_{i,\eta _{i}}^{c_i}(s)-w\right|<\epsilon \right)\text{ d}\eta _{i}\left(s\right). \end{aligned}$$

Note that the distribution of the latter is in general different from that of the chronological local time of $J_{i,\eta _{i}}^{c_i}$, which ultimately implies that the IV t statistic with usual standard errors is not pivotal under nonstationary volatility; see the proof of Proposition 2.

We are now in the position to formulate the following

Lemma 2

Let $f\left(s\right)$ be a continuous function with $\int _{-\infty }^{\infty }\left|f\left(s\right)\right|\text{ d}s<\infty$ and $\int _{-\infty }^{\infty }f\left(s\right)\text{ d}s\ne 0.$ Under Assumptions 1, 2 and 3, we have for $\phi _{i}=-\frac{c_{i}}{T},$ $c_{i}\ge 0,$ as $T\rightarrow \infty$ that

• 1.
  $$\begin{aligned} \frac{1}{T^{0.5+\alpha }}\sum _{t=p+2}^{T}f\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\overset{d}{\rightarrow } {\left\{ \begin{array}{ll} \frac{A_{i}\left(1\right)}{\overline{\omega }_{i}}\left(\int _{-\infty }^{\infty }f\left(s\right)\text{ d}s\right)L_{i}\left(0,1\right) \\ \quad \text{ for}\quad 0\le \alpha <0.5 \\ \int _{0}^{1}f\left(\frac{\overline{\omega }_{i}}{A_{i}\left(1\right)}J_{i,\eta _{i}}^{c_i}(s)\right)\text{ d}s \\ \quad \text{ for}\quad \alpha =0.5\end{array}\right.} \end{aligned}$$
• 2.
  $$\begin{aligned} \frac{1}{T^{0.75+\frac{\alpha }{2}}}\sum _{t=p+2}^{T}\Delta y_{i,t-j}f\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)=o_{p}\left(1\right) \end{aligned}$$

  for $j=1,\ldots ,p$

• 3.
  $$\begin{aligned} \frac{1}{T^{0.5+\alpha }}\sum _{t=p+2}^{T}F^{2}\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\widehat{\varepsilon }_{i,t}^{2}\overset{d}{\rightarrow } {\left\{ \begin{array}{ll} A_{i}\left(1\right)\left(\int _{-\infty }^{\infty }F^{2}\left(s\right)\text{ d}s\right)L_{i}^{*}\left(0,1\right) \\ \quad \text{ for}\quad 0\le \alpha <0.5\ \overline{\omega }^{2} \\ \int _{0}^{1}F^{2}\left(\frac{\overline{\omega }_{i}}{A_{i}\left(1\right)}J_{i,\eta _{i}}^{c_i}(s)\right)\text{ d}\eta _{i}\left(s\right) \\ \quad \text{ for}\quad \alpha =0.5\end{array}\right.} \end{aligned}$$
• 4.
  $$\begin{aligned} \frac{1}{T^{0.25+\frac{\alpha }{2}}}\sum _{t=p+2}^{T}F\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\varepsilon _{i,t}\overset{d}{\rightarrow }{\left\{ \begin{array}{ll} \sqrt{A_{i}\left(1\right)\left(\int _{-\infty }^{\infty }F^{2}\left(s\right)\text{ d}s\right)L_{i}^{*}\left(0,1\right)}\cdot U_{i}\left(1\right) \\ \quad \text{ for}\quad 0\le \alpha <0.5 \\ \overline{\omega }\int _{0}^{1}F\left(\frac{\overline{\omega }_{i}}{A_{i}\left(1\right)}J_{i,\eta _{i}}^{c_i}(s)\right)\text{ d}W\left(\eta _{i}\left(s\right)\right) \\ \quad \text{ for}\quad \alpha =0.5,\end{array}\right.} \end{aligned}$$

where $U_{i}(1)\sim \mathcal N (0,1)$ is independent of $J_{i,\eta _{i}}^{c_i}$ and thus of $L_{i}^{*}\left(0,1\right).$

Note that $F^{2}\left(s\right), sF\left(s\right), sF^{2}\left(s\right)$ and $s^{2}F^{2}\left(s\right)$ all fulfill the assumptions of the lemma, so item 2 applies for $F^{2}\left(T^{-\alpha }y_{i,t-1}\right),$ $T^{-\alpha }y_{i,t-1}F\left(T^{-\alpha }y_{i,t-1}\right), T^{-\alpha }y_{i,t-1}F^{2}$ $\left(T^{-\alpha }y_{i,t-1}\right)$ as well as $T^{-2\alpha }y_{i,t-1}^{2}F\left(T^{-\alpha }y_{i,t-1}\right)$ leading to

$$\begin{aligned} \frac{1}{T^{0.5+\alpha }}\sum _{t=p+2}^{T}F^{2}\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\overset{d}{\rightarrow }{\left\{ \begin{array}{ll} \frac{A_{i}\left(1\right)}{\overline{\omega }_{i}}\left(\int _{-\infty }^{\infty }F^{2}\left(s\right)\text{ d}s\right)L_{i}\left(0,1\right) \\ \quad \text{ for}\quad 0\le \alpha <0.5\ \\ \int _{0}^{1}F^{2}\left(\frac{\overline{\omega }_{i}}{A_{i}\left(1\right)}J_{i,\eta _{i}}^{c_i}(s)\right)\text{ d}s \\ \quad \text{ for}\quad \alpha =0.5\end{array}\right.}, \end{aligned}$$

$$\begin{aligned} \frac{1}{T^{0.5+2\alpha }}\sum _{t=p+2}^{T}y_{i,t-1}F\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\overset{d}{\rightarrow }{\left\{ \begin{array}{ll} \frac{A_{i}^{2}\left(1\right)}{\overline{\omega }_{i}^{2}}\left(\int _{-\infty }^{\infty }sF\left(s\right)\text{ d}s\right)L_{i}\left(0,1\right) \\ \quad \text{ for}\quad 0\le \alpha <0.5\ \\ \frac{\overline{\omega }_{i}}{A_{i}\left(1\right)}\int _{0}^{1}J_{i,\eta _{i}}^{c_i}(s)F\left(\frac{\overline{\omega }_{i}}{A_{i}\left(1\right)} J_{i,\eta _{i}}^{c_i}(s)\right)\text{ d}s \\ \quad \text{ for}\quad \alpha =0.5\end{array}\right.}, \end{aligned}$$

$$\begin{aligned} \frac{1}{T^{0.5+2\alpha }}\sum _{t=p+2}^{T}y_{i,t-1}F^{2}\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\overset{d}{\rightarrow }{\left\{ \begin{array}{ll} \frac{A_{i}^{2}\left(1\right)}{\overline{\omega }_{i}^{2}}\left(\int _{-\infty }^{\infty }sF^{2}\left(s\right)\text{ d}s\right)L_{i}\left(0,1\right) \\ \quad \text{ for}\quad 0\le \alpha <0.5\ \\ \frac{\overline{\omega }_{i}}{A_{i}\left(1\right)}\int _{0}^{1}J_{i,\eta _{i}}^{c_i}(s) F^{2}\left(\frac{\overline{\omega }_{i}}{A_{i}\left(1\right)}J_{i,\eta _{i}}^{c_i}(s) \right)\text{ d}s \\ \quad \text{ for}\quad \alpha =0.5\end{array}\right.}, \end{aligned}$$

and

$$\begin{aligned} \frac{1}{T^{0.5+3\alpha }}\sum _{t=p+2}^{T}y_{i,t-1}^{2}F^{2}\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\overset{d}{\rightarrow }{\left\{ \begin{array}{ll} \frac{A_{i}^{2}\left(1\right)}{\overline{\omega }_{i}^{2}}\left(\int _{-\infty }^{\infty }s^{2}F^{2}\left(s\right)\text{ d}s\right)L_{i}\left(0,1\right) \\ \quad \text{ for}\quad 0\le \alpha <0.5\ \\ \frac{\overline{\omega }_{i}^{2}}{A_{i}^{2}\left(1\right)}\int _{0}^{1}\left(J_{i,\eta _{i}}^{c_i}(s)\right)^{2}F^{2}\left(\frac{\overline{\omega }_{i}}{A_{i}\left(1\right)} J_{i,\eta _{i}}^{c_i}(s)\right)\text{ d}s \\ \quad \text{ for}\quad \alpha =0.5\end{array}\right.}. \end{aligned}$$

Moreover, item 2 applies just as well for $T^{-0.75-\frac{\alpha }{2}}\sum _{t=p+2}^{T}\Delta y_{i,t-j}F\left(T^{-\alpha }y_{i,t-1}\right)$ and $T^{-0.75-\frac{\alpha }{2}}\sum _{t=p+2}^{T}\Delta y_{i,t-j}F^{2}\left(T^{-\alpha }y_{i,t-1}\right),$ and also leads to

$$\begin{aligned} \frac{1}{T^{0.75+3\frac{\alpha }{2}}}\sum _{t=p+2}^{T}\Delta y_{i,t-j}y_{i,t-1}F^{2}\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)=o_{p}\left(1\right). \end{aligned}$$

Proof of Lemma 2

Let $C$ be a generic positive constant.

• 1. For $0\le \alpha <0.5,$ the result follows from Wang and Phillips (2009); see also Park and Phillips (1999), de Jong and Wang (2005). For $\alpha =0.5,$ Lemma 1 and an application of the continuous mapping theorem lead to the desired limit.

• 2. For $0\le \alpha <0.25,$ the proof is a straightforward extension of (Chang et al (2001), Lemma 5e) by picking their $\delta$ such that $\frac{1}{16}<\delta <\frac{1}{8}$. For $0.25\le \alpha \le 0.5,$ write for $h_{T}=C\, T^{\beta }$ with $0<\beta <0.25$

  $$\begin{aligned} \frac{1}{T^{0.75+\frac{\alpha }{2}}}\sum _{t=p+2}^{T}\Delta y_{i,t-j}f\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)&= \frac{1}{T^{0.75+\alpha /2}}\Biggl [\sum _{t=p+2}^{T}\Delta y_{i,t-j}f\left(\frac{y_{i,t-h_{T}}}{T^{\alpha }}\right)\\&\!+\!\sum _{t=p+2}^{T}\Delta y_{i,t-j}\left(f\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\!-\!f\left(\frac{y_{i,t-h_{T}}}{T^{\alpha }}\right)\right)\Biggr ]. \end{aligned}$$

  Using the Cauchy-Schwarz inequality,

  $$\begin{aligned}&\left|\frac{1}{T^{0.75+\frac{\alpha }{2}}}\sum _{t=p+2}^{T}\Delta y_{i,t-j}\left(f\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)-f\left(\frac{y_{i,t-h_{T}}}{T^{\alpha }}\right)\right)\right|\\&\qquad \le \sqrt{\widehat{\sigma }_{\Delta y}^{2}\frac{1}{T^{0.5+\alpha }}\sum _{t=p+2}^{T}\left(f\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)-f\left(\frac{y_{i,t-h_{T}}}{T^{\alpha }}\right)\right)^{2}} \end{aligned}$$

  with $\widehat{\sigma }_{\Delta y}^{2}\!\!=\!\!\frac{1}{T}\sum _{t=p+2}^{T}\left(\Delta y_{i,t-j}\right)^{2}\!\!=\!\!O_{p}\left(1\right).$ The boundedness of the derivative of $f\left(\cdot \right)$ implies that $\left|f\left(T^{-\alpha }y_{i,t-1}\right)\!\!-\!\!f\left(T^{-\alpha }y_{i,t-h_{T}}\right)\right|^{2}\!\le \! CT^{-2\alpha }\left|\sum _{j=t-h_{T}}^{t-1}\Delta y_{i,t-j}\right|^{2}$ so, noting that $\sum _{j=t-h_{T}}^{t-1}\Delta y_{i,t-j}=O_{p}\bigl (h_{T}^{1/2}\bigr ),$ we obtain

  $$\begin{aligned} \frac{1}{T^{0.5+\alpha }}\sum _{t=p+2}^{T}\left(f\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)-f\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\right)^{2}&= O_{p}\left(h_{T}T^{0.5-3\alpha }\right)\\&= O_{p}\left(T^{0.5+\beta -3\alpha }\right).\\&= o_{p}\left(1\right). \end{aligned}$$

  With $\Delta y_{t-j}$ composed additively of a linear process with exponentially decaying coefficients and a negligible nonstationary component, we have that

  $$\begin{aligned} \Delta y_{i,t-j}f\left(\frac{y_{i,t-h_{T}}}{T^{\alpha }}\right)&= \sum _{k=0}^{h_{T}-j-1}b_{k}f\left(\frac{y_{i,t-h_{T}}}{T^{\alpha }}\right)\varepsilon _{t-k-j}\\&\quad +\,\sum _{k\ge h_{T}-j}b_{k}f\left(\frac{y_{i,t-h_{T}}}{T^{\alpha }}\right)\varepsilon _{t-k-j}+o_{p}\left(T^{-0.5}\right); \end{aligned}$$

  the first term on the r.h.s. is a linear combination of bounded-variance martingale differences, while the second has absolute expectation of order $\sum _{k\ge h-j}\left|b_{k}\right|;$ rearranging the terms when summing over $t$ leads to the desired result.

• 3. We have that

  $$\begin{aligned}&\frac{1}{T^{0.5+\alpha }}\sum _{t=p+2}^{T}F^{2}\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\widehat{\varepsilon }_{i,t}^{2}=\frac{1}{T^{0.5+\alpha }}\sum _{t=p+2}^{T}F^{2}\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\varepsilon _{i,t}^{2}\\&\quad \quad \quad -\frac{2}{T^{0.5+\alpha }}\sum _{t=p+2}^{T}F^{2}\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\left(\widehat{\phi }_{i}^{iv}y_{i,t-1}+\sum _{j=1}^{p}\left(\widehat{a}_{ij}-a_{ij}\right)\Delta y_{i,t-j}\right)\\&\quad \quad \quad +\frac{1}{T^{0.5+\alpha }}\sum _{t=p+2}^{T}F^{2}\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\left(\widehat{\phi }_{i}^{iv}y_{i,t-1}+\sum _{j=1}^{p}\left(\widehat{a}_{ij}-a_{ij}\right)\Delta y_{i,t-j}\right)^{2}. \end{aligned}$$

  Write for the first term on the r.h.s.

  $$\begin{aligned} \frac{1}{T^{0.5+\alpha }}\sum _{t=p+2}^{T}F^{2}\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\varepsilon _{i,t}^{2}=\frac{1}{T^{0.5+\alpha }}\sum _{t=p+2}^{T}F^{2}\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\sigma _{i,t}^{2}+o_{p}\left(1\right) \end{aligned}$$

  since $F^{2}\left(y_{i,t-1}\right)\left(\varepsilon _{i,t}^{2}-\sigma _{i,t}^{2}\right)$ is by construction a martingale difference sequence so that $\mathrm Var \left(T^{-0.5-\alpha }\sum _{t=p+2}^{T}F^{2}\left(T^{-\alpha }y_{i,t-1}\right)\left(\varepsilon _{i,t}^{2}-\sigma _{i,t}^{2}\right)\right)$ is given by

  $$\begin{aligned} \frac{1}{T^{1+2\alpha }}\sum _{t=p+2}^{T}\mathrm Var \left(F^{2}\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\right)\mathrm Var \left(\left(\varepsilon _{i,t}^{2}-\sigma _{i,t}^{2}\right)\right) \end{aligned}$$

  Since $y_{i,t-1}\!=\!O_p(T^{0.5})$ and using integrability of $F$, we have $\mathrm Var \left(F^{2}\left(T^{-\alpha }y_{i,t-1}\right)\right)$ $\rightarrow 0$ for $0\le \alpha <0.5$. For $\alpha =0.5$, $\frac{y_{i,t-1}}{T^{\alpha }}=O_p(1)$. Assumption 3 implies that $F$ is bounded, whence $\mathrm Var \left(F^{2}\left(T^{-0.5}y_{i,t-1}\right)\right)=O(1)$ such that

  $$\begin{aligned} \frac{1}{T^{2}}\sum _{t=p+2}^{T}\mathrm Var \left(F^{2}\left(\frac{y_{i,t-1}}{T^{0.5}}\right)\right)\mathrm Var \left(\left(\varepsilon _{i,t}^{2}-\sigma _{i,t}^{2}\right)\right)=O(T^{-1}). \end{aligned}$$

  For the second term, we have with Lemma 2 that

  $$\begin{aligned}&\frac{2\widehat{\phi }_{i}^{iv}}{T^{0.5+\alpha }} \sum _{t=p+2}^{T}F^{2}\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)y_{i,t-1} \\&+\frac{2}{T^{0.5+\alpha }}\sum _{j=1}^{p}\left(\widehat{a}_{ij}-a_{ij}\right)\sum _{t=p+2}^{T}F^{2}\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\Delta y_{i,t-j} \end{aligned}$$

  is $o_{p}\left(1\right)$, since $\widehat{\phi }_{i}^{iv}=O_{p}\left(T^{-0.25-3\frac{\alpha }{2}}\right)$ and $\widehat{a}_{ij}-a_{ij}=O_{p}\left(T^{-0.25}\right)$. Next, the Cauchy-Schwarz inequality implies that $\sum _{t=p+2}^{T}F^{2}\left(T^{-\alpha }y_{i,t-1}\right)\left(\Delta y_{i,t-j}\right)^{2}=O_{p}\left(T^{0.75+\alpha }\right)$. Then, similar arguments apply for the third term. The result follows for $0\le \alpha <0.5$ with

  $$\begin{aligned} \frac{1}{T^{0.5+\alpha }}\sum _{t=p+2}^{T}F^{2}\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\sigma _{i,t}^{2}\overset{d}{\rightarrow }L_{i}^{*}\left(1,0\right), \end{aligned}$$

  (3)

  which is established with a tedious, yet straightforward adaptation of the arguments of Wang and Phillips (2009). For $\alpha =0.5,$ the continuous mapping theorem delivers

  $$\begin{aligned} \frac{1}{T}\sum _{t=p+2}^{T}F^{2}\left(\frac{y_{i,t-1}}{\sqrt{T}}\right)\sigma _{i,t}^{2}\overset{d}{\rightarrow }\overline{\omega }_{i}^{2}\int _{0}^{1}F^{2}\left(\frac{\overline{\omega }_{i}}{A_{i}\left(1\right)} J_{i,\eta _{i}}^{c_i}(s)\right)\text{ d}\eta _{i}\left(s\right) \end{aligned}$$

  as required.

• 4. The result follows for $0\le \alpha <0.5$ with Equation (3) and the arguments used by (Phillips et al (2004), Lemma 3.1); for $\alpha =0.5,$ use Theorem 2.1 in Hansen (1992).

Proof of Proposition 1

With Lemma 2 item 2, the stationarity of $\mathbf x _{t}$ and the md property of $\mathbf x _{t-1}\varepsilon _{t}$, we obtain immediately that

$$\begin{aligned} A_{i,T}=\sum _{t=p+2}^{T}F\left(T^{-\alpha }y_{i,t-1}\right)\varepsilon _{i,t}+o_{p}\left(T^{0.25+{\alpha }/{2}}\right); \end{aligned}$$

with $T^{-1}\sum _{t=p+2}^{T}\Delta y_{i,t-j}y_{i,t-1}=O_{p}\left(1\right)$ it follows that

$$\begin{aligned} B_{i,T}=\sum _{t=p+2}^{T}F\left(T^{-\alpha }y_{i,t-1}\right)y_{i,t-1}+o_{p}\left(T^{0.75+\alpha /2}\right). \end{aligned}$$

Since $\sum _{t=p+2}^{T}F\left(T^{-\alpha }y_{i,t-1}\right)y_{i,t-1}$ is of exact order $\Theta _{p}\left(T^{0.5+2\alpha }\right),$ $B_{i,T}$ is at least of magnitude $O_{p}\left(T^{0.5+2\alpha }\right),$ implying the desired convergence rate for $\widehat{\phi }_{i}^{iv}.$ Similar arguments lead to the desired consistency of $\widehat{\mathbf{a }}_{i}.$

Proof of Proposition 2

We build on the behavior of $A_{i,T}$ from the proof of Proposition 1 for $c_{i}=0$. Note that

$$\begin{aligned} C_{i,T}=\sum _{t=p+2}^{T}F^{2}\left(T^{-\alpha }y_{i,t-1}\right)\widehat{\varepsilon }_{i,t}^{2}+o_{p}\left(T^{0.5+\alpha }\right) \end{aligned}$$

Lemma 2 items 3 and 4 implies that $C_{i,T}$ consistently estimates $\mathrm Var (A_{i,T})$, which, since $t_i^{rob}=A_{i,T}/\sqrt{C_{i,T}}$ under the null, completes the proof.

With usual standard errors, $C_{i,T}$ reduces to $\widehat{\sigma }_{i}^{2}\sum _{t=p+2}^{T}F^{2}\left(T^{-\alpha }y_{i,t-1}\right)+o_{p}\left(T^{0.5+\alpha }\right)$ which, properly normalized, has the wrong weak limit, and the resulting t statistic is not pivotal even if $0\le \alpha <0.5;$ for $\alpha =0.5,$ it is straightforward to check that the limit is indeed the same if using Eicker-White or usual standard errors, but the resulting distribution depends on $\eta _{i}\left(s\right)$.

Proof of Proposition 3

Using Lemma 2, we obtain that

$$\begin{aligned} \frac{A_{i,T}}{T^{0.25+\frac{\alpha }{2}}}&= \frac{1}{T^{0.25+\frac{\alpha }{2}}}\sum _{t=p+2}^{T}F\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)\varepsilon _{i,t}\\&\quad -\,\frac{A_{i}\left(1\right)c_{i}}{T^{1.25+\frac{\alpha }{2}}}\sum _{t=p+2}^{T}y_{i,t-1}F\left(\frac{y_{i,t-1}}{T^{\alpha }}\right)+o_{p}\left(T^{0.25+\frac{\alpha }{2}}\right); \end{aligned}$$

the second term on the r.h.s. is of exact order of magnitude $\Theta _{p}\left(T^{-0.75+3\frac{\alpha }{2}}\right)$ and has a nondegenerate limit only for $\alpha =0.5.$ The result follows analogously to the proof of Proposition 2 with

$$\begin{aligned} C_{i,T}=\sum _{t=p+2}^{T}F^{2}\left(T^{-\alpha }y_{i,t-1}\right)\widehat{\varepsilon }_{i,t}^{2}+o_{p}\left(T^{0.5+\alpha }\right) \end{aligned}$$

and Lemma 2 items 3 and 4.

Proof of Proposition 4

Use a Taylor expansion, $F\left(T^{-\alpha }y_{i,t-1}\right)=F^{\prime }\left(0\right)\cdot T^{-\alpha }y_{i,t-1}+o_{p}\left(T^{-2\alpha }\right)$; the result follows in a straightforward manner.

Proof of Proposition 5

Considering the arguments of the proof of Proposition 3 in Demetrescu et al (2011), we only need to show that

$$\begin{aligned} \int _{\text{0}}^{1}F_{1}\left(\frac{\sqrt{T}}{T^{\alpha }}B_{1}\left(s\right)\right)F_{2}\left(\frac{\sqrt{T}}{T^{\alpha }}B_{2}\left(s\right)\right)\text{ d}s=o_{p}\left(\frac{1}{\sqrt{T}}\right) \end{aligned}$$

(4)

where $B_{i}$ are two independent, standard Wiener processes. The invariance properties of Wiener processes lead immediately to

$$\begin{aligned} \int _{\text{0}}^{1}F_{1}\left(\frac{\sqrt{T}}{T^{\alpha }}B_{1}\left(s\right)\right)F_{2}\left(\frac{\sqrt{T}}{T^{\alpha }}B_{2}\left(s\right)\right)\text{ d}s \overset{d}{=}\int _{\text{0}}^{1}F_{1}\left(B_{1}\left(T^{1-2\alpha }s\right)\right)F_{2}\left(B_{2}\left(T^{1-2\alpha }s\right)\right)\text{ d}s \end{aligned}$$

with “$\overset{d}{=}$” standing for equivalence in distribution. Consider the change of variable $t=T^{1-2\alpha }s$, leading to

$$\begin{aligned} \int _{\text{0}}^{1}F_{1}\left(\frac{\sqrt{T}}{T^{\alpha }}B_{1}\left(s\right)\right)F_{2}\left(\frac{\sqrt{T}}{T^{\alpha }}B_{2}\left(s\right)\right)\text{ d}s \overset{d}{=}\int _{\text{0}}^{\frac{1}{T^{1-2\alpha }}}F_{1}\left(B_{1}\left(t\right)\right)F_{2}\left(B_{2}\left(t\right)\right)\text{ d}t. \end{aligned}$$

To complete the proof, recall from Kallianpur and Robbins (1953) that the integral on the r.h.s. is of order $O_{p}\left(T^{-1+2\alpha }\log T^{1-2\alpha }\right)=O_{p}\left(T^{-1+2\alpha }\log T\right);$ the desired result (4) follows when $\alpha <0.375.$