# HAR Testing for Spurious Regression in Trend

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## Abstract

**:**

It is meaningless to talk about ‘confirming’ theories when spurious results are so easily obtained.

## 1. Introduction

## 2. Regression of Stochastic Trend on Time Polynomials

#### 2.1. Model Details and Background

#### 2.2. Three t-Statistics

**Theorem**

**1.**

**Remark**

**1.**

**Theorem**

**2.**

**Remark**

**2.**

## 3. Regressions Among Independent Random Walks

**Theorem**

**3.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

## 4. Simulations

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

${L}_{2}\left(\right)open="["\; close="]">0,1$ | space of square integrable functions on $\left(\right)$. |

⟹ | weak convergence. |

$\u230a\xb7\u230b$ | integer part of. |

$:=$ | definitional equality. |

${o}_{p}\left(1\right)$ | tends to zero in probability. |

${o}_{a.s}\left(1\right)$ | tends to zero almost surely. |

${O}_{p}\left(1\right)$ | bounded in probability. |

$\stackrel{p}{\u27f6}$ | converge in probability. |

$r\wedge s$ | min$\left(\right)$. |

∼ | asymptotic equivalence. |

≡ | distributional equivalence. |

${\sim}_{d}$ | distributed as |

## Appendix A. Proofs of Theorems in Section 2

**Lemma**

**A1.**

- (i)
- $\phantom{\rule{4pt}{0ex}}{\underline{B}}_{{\phi}_{K}}\left(r\right)\sim {O}_{p}\left(\right)open="("\; close=")">1/\sqrt{K}$ uniformly in $r\in \left(\right)open="["\; close="]">0,1$,
- (ii)
- $K{\int}_{0}^{1}{\underline{B}}_{{\phi}_{K}}^{2}={\omega}^{2}/{\pi}^{2}+{o}_{p}\left(1\right)$ with ${\omega}^{2}=2\pi {f}_{\mu}\left(0\right)$,
- (iii)
- $\phantom{\rule{4pt}{0ex}}{\int}_{0}^{1}{\underline{B}}_{{\phi}_{K}}^{2}{\left(\right)}^{{C}_{K}^{\prime}}2$,
- (iv)
- ${\int}_{0}^{1}{\int}_{0}^{1}{k}_{b}\left(\right)open="("\; close=")">r-q\left(\right)open="["\; close="]">{C}_{K}^{\prime}{\overline{\phi}}_{K}\left(q\right)$

**Lemma**

**A2.**

- (i)
- ${C}_{K}^{\prime}{\widehat{\beta}}_{K}/\sqrt{n}={C}_{K}^{\prime}{\widehat{\alpha}}_{K}={C}_{K}^{\prime}{Z}_{K}+{o}_{p}\left(1\right),$
- (ii)
- $K\left(\right)open="("\; close=")">{s}_{b}^{2}{C}_{K}^{\prime}{\left(\right)}^{{\mathsf{\Phi}}_{K}^{\prime}}-1=K{\int}_{0}^{1}{\underline{B}}_{{\phi}_{K}}^{2}+{o}_{p}\left(1\right),$
- (iii)
- $\frac{K}{M}\left(\right)open="("\; close=")">{\widehat{\omega}}_{{C}_{K}^{\prime}{\beta}_{K}}^{2}K{\int}_{0}^{1}{\underline{B}}_{{\phi}_{K}}^{2}{\left(\right)}^{{C}_{K}^{\prime}}2$
- (iv)
- $\frac{K}{n}\left(\right)open="("\; close=")">{\stackrel{\u02c7}{\omega}}_{{C}_{K}^{\prime}{\beta}_{K}}^{2}{\underline{B}}_{{\phi}_{K}}\left(r\right){\underline{B}}_{{\phi}_{K}}\left(q\right)\left(\right)open="["\; close="]">{C}_{K}^{\prime}{\overline{\phi}}_{K}\left(r\right)$where ${Z}_{K}={\left(\right)}_{{z}_{k}}^{}k=1K$ are the random coefficients in the orthonormal representation (5), ${s}_{b}^{2}$, ${\widehat{\omega}}_{{C}_{K}^{\prime}{\beta}_{K}}^{2}$ and ${\stackrel{\u02c7}{\omega}}_{{C}_{K}^{\prime}{\beta}_{K}}^{2}$ are defined as in formulae (8), (10) and (13), respectively.

**Proof**

**of**

**Lemma**

**A1.**

**Proof**

**of**

**Lemma**

**A2.**

**Proof**

**of**

**Theorem**

**1.**

## Appendix B. Derivations Leading to (23)–(28)

**Lemma**

**A3.**

- (i)
- for $r\in \left(\right)open="["\; close="]">0,1$,$$\frac{{\widehat{u}}_{\left(\right)}}{}\sqrt{n}r:=\underline{B}\left(r\right)\phantom{\rule{4.pt}{0ex}};$$
- (ii)
- $${n}^{2}{\left(\right)}^{{s}_{a}}2$$
- (iii)
- $$\frac{{n}^{2}}{M}{\left(\right)}^{{\widehat{\omega}}_{a}}2$$
- (iv)
- $$n{\left(\right)}^{{\stackrel{\u02c7}{\omega}}_{a}}2\underline{B}\left(r\right)\underline{B}\left(q\right)rqdrdq\phantom{\rule{4.pt}{0ex}};$$

**Proof**

**of**

**Lemma**

**A3.**

**Proof**

**of**

**(23)–(28).**

## Appendix C. Proof of the Theorem in Section 3

**Proof**

**of**

**Theorem**

**3.**

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1. | Heteroskedastic robust standard errors were introduced by (Eicker 1967; Huber 1967; White 1980). HAC estimators were introduced by (White 1982) and have a long subsequent history of enhancement. |

2. | Heteroskedastic and autocorrelation robust standard errors were introduced in (Kiefer and Vogelsang 2002a, 2002b) and, following this lead, (Phillips 2005) used the HAR terminology to characterize a class of robust inferential procedures in an article concerned with the development of automated mechanisms of valid inference in econometrics. Other important early contributions concerning HAC covariance matrix estimators without truncation were given by (Kiefer and Vogelsang 2005; Kiefer et al. 2000; Robinson 1998). |

3. | Weakly dependent innovations in the form of an AR(1) error process, viz., ${\mu}_{s}=\rho {\mu}_{s-1}+{\epsilon}_{s}$, with ${\epsilon}_{s}{\sim}_{d}iid$ $N\left(\right)open="("\; close=")">0,1$ were also considered. The results were similar and so only the $iid$ case is reported here. |

**Figure 2.**Rejection frequencies of the usual t and heteroskedasticity and autocorrelation consistent (HAC) t-statistics in spurious trend regression of a random walk calculated based on 10,000 simulations with sample size $n=200$ and the critical values from the standard Normal distribution at $5\%$ significance level.

**Figure 3.**Rejection frequencies of the heteroskedasticity and autocorrelation robust (HAR) t-statistic in spurious trending regressions.

**Figure 7.**Rejection frequencies of the usual t and HAC t-statistics in spurious regressions among random walks calculated based on 10,000 simulations with sample size $n=200$ and critical values from the standard Normal distribution at $5\%$ significance level.

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**MDPI and ACS Style**

Phillips, P.C.B.; Wang, X.; Zhang, Y.
HAR Testing for Spurious Regression in Trend. *Econometrics* **2019**, *7*, 50.
https://doi.org/10.3390/econometrics7040050

**AMA Style**

Phillips PCB, Wang X, Zhang Y.
HAR Testing for Spurious Regression in Trend. *Econometrics*. 2019; 7(4):50.
https://doi.org/10.3390/econometrics7040050

**Chicago/Turabian Style**

Phillips, Peter C. B., Xiaohu Wang, and Yonghui Zhang.
2019. "HAR Testing for Spurious Regression in Trend" *Econometrics* 7, no. 4: 50.
https://doi.org/10.3390/econometrics7040050