Detecting multiple breaks in long memory the case of U.S. inflation

Abstract

Multiple structural change tests by Bai and Perron (Econometrica 66:47–78, 1998) are applied to the regression by Demetrescu et al. (Econ Theory 24:176–215, 2008) in order to detect breaks in the order of fractional integration. With this instrument we tackle time-varying inflation persistence as an important issue for monetary policy. We determine not only the location and significance of breaks in persistence, but also the number of breaks. Only one significant break in U.S. inflation persistence (measured by the long-memory parameter) is found to have taken place in 1973, while a second break in 1980 is not significant.

Appendix

1.1 Proof of Proposition 1

Under Gaussianity and Assumption 2 the pseudo-log-likelihood function becomes

$$\begin{aligned} \mathcal L (\theta _1, \ldots , \theta _m ; \, d,\sigma ^2)=-\frac{T}{2} \log (2\pi \sigma ^2)-\frac{1}{2\sigma ^2}\sum _{t=1}^{T}\varepsilon _t^2, \end{aligned}$$

with $\varepsilon _t = \Delta ^{d+\theta _{j-1}} y_t$ for $t=T_{j-1}+1, \ldots , T_j$ using (3) and (4), or $\varepsilon _t = \Delta ^{d+\theta _{j}} y_t$ for $t=T_{j}+1,\ldots ,T_{j+1}$, such that ($j=1,\ldots ,m$)

$$\begin{aligned} \frac{\partial \varepsilon _t}{\partial \theta _j}= \left\{ \begin{array}{cl} \left( \log (1-L)\right) (1-L)^{d+\theta _j}y_t, &{} t=T_{j}+1,\ldots ,T_{j+1} \\ 0 &{} \text{ else } \end{array} \right. . \end{aligned}$$

With $\log (1-L)=-\sum _{j=1}^{\infty }j^{-1}L^{j}$ we obtain for the score vector evaluated under the null (where $\Delta ^d y_t = \varepsilon _t$)

$$\begin{aligned} S= \left( \left. \frac{\partial \mathcal L }{\partial \theta _j} \right| _{\theta _j=0} \right) _{j=1,\ldots ,m} \!=\! \left( \frac{1}{\sigma ^2} \sum _{t=T_j+1}^{T_{j+1}} \varepsilon _t\varepsilon _{t-1}^{**}\right) _{j=1,\ldots ,m} \quad \text{ with } \varepsilon _{t-1}^{**}\!=\!\sum _{j=1}^{\infty }j^{-1}\varepsilon _{t-j}, \end{aligned}$$

where $\{\varepsilon _{t-1}^{**}\}$ is a stationary process with variance

$$\begin{aligned} \sigma _{**}^2 = \text{ Var } \left( \varepsilon _{t-1}^{**}\right) = \sigma ^2 \sum _{j=1}^\infty j^{-2} = \sigma ^2 \frac{\pi ^2}{6}. \end{aligned}$$

To construct the LM statistic we compute the Fisher information as the outer product of gradients,

$$\begin{aligned} \mathcal I = \text{ E } \left( S \, S^\prime \right) = \frac{\sigma _{**}^2}{ \sigma ^2} \, \text{ diag } \left( T_2-T_1, \ldots , T_{m+1}- T_m \right) . \end{aligned}$$

Hence, we obtain

$$\begin{aligned} S^\prime \mathcal I ^{-1} S = \frac{1}{\sigma _{**}^2 \sigma ^2} \ \sum _{j=1}^m \ \frac{\left( \sum \limits _{t = T_j +1}^{T_{j+1}} \varepsilon _t \varepsilon _{t-1}^{**} \right) ^2}{T_{j+1} - T_j}. \end{aligned}$$

Since the LM statistic is evaluated under $H_0$, we replace $\varepsilon _t$ with $\Delta ^d y_t$. Given the starting value assumption in Assumption 2, this coincides with $x_t$ defined in (7). Consequently, $\varepsilon _{t-1}^{**}$ equals $x_{t-1}^*$ from (9 ), and the LM statistic becomes $LM$ from (8) as required. $\square$

1.2 Proof of Proposition 2

Write the regression equation (13) in obvious matrix notation, $y = X \, \widehat{\beta }+ \widehat{\varepsilon }$, with

$$\begin{aligned} y^\prime&= (x_2,\ldots ,x_T), \\ \widehat{\beta }^\prime&= (\widehat{\phi }, \, \widehat{\psi }_1,\ldots ,\widehat{\psi }_m), \end{aligned}$$

and $X$ containing $(x_1^*,\ldots ,x_{T-1}^*)^\prime$ as the first column, while the other columns contain zeros and segments of $(x_{T_j}^*,\ldots ,x_{T_{j+1}-1}^*)^\prime$. Under Assumption 2, we have $x_{t}=\varepsilon _{t} \sim iid (0,\sigma ^{2})$. The required limiting distributions can be obtained as set out by Robinson (1991) or Tanaka (1999), see also Hassler and Breitung (2006, Lemma A):

$$\begin{aligned} \frac{X^\prime X}{T} \ \overset{p}{\rightarrow } \ \sigma ^{2} \, \frac{\pi ^{2}}{6} \, \Lambda ^0 \end{aligned}$$

where $\overset{p}{\rightarrow }$ stands for convergence in probability, and

$$\begin{aligned} \Lambda ^0 \!=\! \left( \begin{array}{ll} 1 &{}\quad (\lambda ^0)^\prime \\ \lambda ^0 &{}\quad \text{ diag } (\lambda _2^0-\lambda _1^0,\ldots ,\lambda _{m+1}^0-\lambda _m^0) \end{array} \right) , \quad (\lambda ^0)^\prime \!=\! (\lambda _2^0\!-\!\lambda _1^0,\ldots ,\lambda _{m+1}^0\!-\!\lambda _m^0), \end{aligned}$$

and

$$\begin{aligned} \frac{X^\prime y}{\sqrt{T}} \ \overset{d}{\rightarrow } \ \mathcal N _{m+1} \left( \left( \begin{array}{ll} 0 &{} \\ \vdots &{} \\ 0 &{} \end{array} \right) , \ \sigma ^{4} \frac{\pi ^{2}}{6} \, \Lambda ^0 \right) . \end{aligned}$$

Consequently, $\sqrt{T} \, \widehat{\beta }$ follows a limiting normal distribution with $\Sigma = \frac{6}{\pi ^{2}} \left( \Lambda ^0\right) ^{-1}$. Define the $m \times (m+1)$ matrix $R$ with $R \, \widehat{\beta }=( \widehat{\psi }_1,\ldots ,\widehat{\psi }_m)^\prime$. The $F$ statistic becomes

$$\begin{aligned} F(\lambda _1^0,\ldots ,\lambda _m^0) = \frac{T-m-1}{m} \, \frac{ \widehat{\beta }^\prime R^\prime \left( R \, (X^\prime X)^{-1} R^\prime \right) ^{-1} R \, \widehat{\beta }}{\widehat{\varepsilon }^\prime \varepsilon }, \end{aligned}$$

and its limiting distribution follows the usual way. $\square$