Spurious Regression Under Broken‐Trend Stationarity

Abstract

We study the phenomenon of spurious regression between two random variables, when the generating mechanism of individual series is assumed to follow a stationary process around a trend with (possibly) multiple breaks in the level and slope of trend. We develop the relevant asymptotic theory and show that the phenomenon of spurious regression occurs independently of the structure assumed for the errors. In contrast to previous findings, the presence of a spurious relationship will be less severe when breaks are present in the generating mechanism of individual series. This is true whether the regression model includes a linear trend or not. Simulations confirm our asymptotic results, and reveal that in finite samples, the phenomenon of spurious regression is sensitive to the presence of a linear trend in the regression model, and to the relative location of breaks within the sample.

Useful conclusions from surprising results

“…(b) Noriega and Ventosa-Santaularia (2006) considered stationary series around linear trends, with multiple breaks and found evidence of spurious relationships, although these were reduced if a linear trend was included in the regression. In a related study the same authors (2007) consider a variety of cases involving I(0), I(1) series with breaks and simple trends and find evidence of spurious relationships in many realistic cases except for samples of size 25.…”

New bootstrap inference for spurious regression problems

“…What about the inference (CI coverage) about the slope coefficient for τ t ? Table 9 considers the case when the focus is on the coverage for the second slope coefficient of the time variable τ t in Equation (10). The coverage probabilities for all cases in the last Four columns exceed 0.95, suggesting that the meboot remains conservative and reliable at the cost of larger widths.…”

“…Noriega and Ventosa-Santaularia [10] discuss asymptotic theory confirming the presence of spurious regression despite 'breaks' in the levels and trends of the series. Since breaks in the data are shown to matter, even asymptotically, our simulation will incorporate artificial breaks via impulse or step modifications to η t before computing their cumulative sums.…”

“…The maximum entropy (ME) principle is used to claim that the ME density is 'uniform'. For the simple example, the ME density is shown to consist of five half open uniform density intervals: U(−11, 6] * U(6, 10] * U (10,16] * U (16,28] * U(28, 51], illustrated in Figure 1.…”

Spurious Instrumental Variables

“…Indeed, the order in probability of tδ Noriega and Ventosa-Santaulària (2006) and Noriega and Ventosa-Santaularia (2007)]. In this paper, we are concerned with the estimation of equation (1) by Instrumental Variables (hereinafter, IV).…”