Testing the order of fractional integration when smooth deterministic trends are possibly present

Abstract

This paper introduces a test for fractional integration in a model that possibly contains smooth deterministic trends. We model the trend component using a Chebyshev polynomial and specify the short-run dynamics semi-parametrically, accommodating a broad class of possibly nonlinear processes, including those with conditional heteroskedasticity. We use a local Whittle approach for constructing a Lagrange multiplier test statistic and for constructing a frequency-domain information criterion for the selection of the order of the Chebyshev polynomial. We show that widely used time-domain information criteria are generally inconsistent for the true order, whereas our frequency-domain criterion remains robust under both short- and long-memory behaviour. Monte Carlo simulations and an empirical application to the UK Great Ratios support our theoretical findings.

Figures (5)

• Figure 1: Simulated rejection frequencies of $H_0 : \delta = 0$, averaged over 5,000 replications. The DGP sets $k_0 = 0$ with $\beta_{0,0} = 0$, while $\epsilon_t \sim \text{NIID}(0,1)$ and $\delta_0 \in [-0.499, 0.499]$. The model is given by $y_t = \beta_0 + \Delta^{-\delta} \epsilon_t$ when $k = 0$, and by $y_t = \beta_0 + \beta_1 P_t(1) + \Delta^{-\delta} \epsilon_t$ when $k = 1$. The LM-statistic is based on the OLS residuals $\hat{u}_t (k)$ in \ref{['res']}.
• Figure 2: Simulated rejection frequencies of $H_0 : \delta = 0$, averaged over 5,000 replications. The DGP sets $k_0 = 1$ with $\beta_{0,0} = 0$ and $\beta_{1,0} = 1$, while $\epsilon_t \sim \text{NIID}(0,1)$ and $\delta_0 \in [-0.499, 0.499]$. The model is given by $y_t = \beta_0 + \Delta^{-\delta} \epsilon_t$ when $k = 0$, and by $y_t = \beta_0 + \beta_1 P_t(1) + \Delta^{-\delta} \epsilon_t$ when $k = 1$. The LM-statistic is based on the OLS residuals $\hat{u}_t (k)$ in \ref{['res']}.
• Figure 3: Simulated rejection frequencies of $H_0 : \delta = 0$, averaged over 5,000 replications. The DGP sets $k_0 = 1$ with $\beta_{0,0} = 0$ and $\beta_{1,0} = 1$, while $\epsilon_t \sim \text{NIID}(0,1)$ and $\delta_0 \in [-0.499, 0.499]$. The model is given by $y_t = \sum_{n = 0}^k \beta_n P_t (n) + \Delta^{-\delta} \epsilon_t$ with the order $k$ selected by either the usual time-domain HQ information criterion, the infeasible $\delta$-HQ criterion or our LW-HQ criterion. The correct specification with $k = k_0$ is included as a benchmark. The LM-statistic is based on the OLS residuals $\hat{u}_t (k)$ in \ref{['res']}. For LW-HQ, the bandwidth is fixed at $m = \left\lfloor T^{0.65} \right\rfloor$.
• Figure 4: Histograms of the estimated Chebyshev polynomial order selected by three HQ-type information criteria. The DGP is as in \ref{['eq1']}--\ref{['eq21']} with $u_t = \Delta^{-\delta_0} \eta_t$, where $\eta_t \sim \text{NIID}(0,1)$ and $\delta_0 \in \{-0.3, 0, 0.35, 0.45\}$. The true Chebyshev polynomial is specified by $k_0 = 1$, $\beta_{0,0} = 0$ and $\beta_{1,0} = 1$. The selection frequencies are averaged across 5,000 replications. The results for the time domain HQ criterion are shown in red, those for the infeasible $\delta$-HQ criterion proposed by lavielle2000least in green, and for our LW-HQ criterion in magenta. For LW-HQ we fix the bandwidth $m = \left\lfloor T^{0.65} \right\rfloor$.
• Figure 5: Logged UK Great Ratios. Dotted lines show the estimated deterministic component $\sum_{n=0}^{\hat{k}}\hat{\beta}_n P_t(n)$ with the order $\hat{k}$ selected by time domain BIC (black) and frequency-domain LW-BIC (red).

Theorems & Definitions (18)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma A.1
• proof
• Lemma A.2
• proof
• Lemma A.3