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The modified conditional sum-of-squares estimator for fractionally integrated models

Mustafa R. Kılınç, Michael Massmann

Abstract

In this paper, we analyse the influence of estimating a constant term on the bias of the conditional sum-of-squares (CSS) estimator in a stationary or non-stationary type-II ARFIMA ($p_1$,$d$,$p_2$) model. We derive expressions for the estimator's bias and show that the leading term can be easily removed by a simple modification of the CSS objective function. We call this new estimator the modified conditional sum-of-squares (MCSS) estimator. We show theoretically and by means of Monte Carlo simulations that its performance relative to that of the CSS estimator is markedly improved even for small sample sizes. Finally, we revisit three classical short datasets that have in the past been described by ARFIMA($p_1$,$d$,$p_2$) models with constant term, namely the post-second World War real GNP data, the extended Nelson-Plosser data, and the Nile data.

The modified conditional sum-of-squares estimator for fractionally integrated models

Abstract

In this paper, we analyse the influence of estimating a constant term on the bias of the conditional sum-of-squares (CSS) estimator in a stationary or non-stationary type-II ARFIMA (,,) model. We derive expressions for the estimator's bias and show that the leading term can be easily removed by a simple modification of the CSS objective function. We call this new estimator the modified conditional sum-of-squares (MCSS) estimator. We show theoretically and by means of Monte Carlo simulations that its performance relative to that of the CSS estimator is markedly improved even for small sample sizes. Finally, we revisit three classical short datasets that have in the past been described by ARFIMA(,,) models with constant term, namely the post-second World War real GNP data, the extended Nelson-Plosser data, and the Nile data.
Paper Structure (39 sections, 47 theorems, 369 equations, 5 figures, 9 tables)

This paper contains 39 sections, 47 theorems, 369 equations, 5 figures, 9 tables.

Table of Contents

  1. Introduction
  2. The modified conditional sum-of-squares estimator
  3. The model
  4. The conditional sum-of-squares estimator
  5. The modified profile likelihood
  6. The modified conditional sum-of-squares estimator
  7. Relationship with other modifications
  8. Asymptotic biases
  9. Special cases
  10. ARFIMA(0,d,0) model
  11. ARFIMA(1,d,0) model
  12. Short-memory models
  13. The role of unobserved pre-sample values
  14. Simulation
  15. Empirical examples
  16. ...and 24 more sections

Key Result

Theorem 2.1

Let $x_t$, $t$ = 1,$\ldots$,$T$, be given by genq1-repmainf and let Assumptions A2 to A5 be satisfied. Then, the expected scores of $L^* (d,\varphi)$, evaluated at the true parameters $d_0$ and $\varphi_0$, are given by when $T \rightarrow \infty$. The expected scores of $L_{\mu_0}^* (d,\varphi)$, evaluated at the true parameters $d_0$ and $\varphi_0$, are given by

Figures (5)

  • Figure 1: Panel (a) plots the modification term $m(d)$ in \ref{['genmodificationterm']} for $d$ between $-1$ and 2, and $T$ = 32, 64, 128, 256, and without short-run dynamics, i.e. $\phi(L;\varphi) = 1$. The value of $d = 1/2$ is added as a vertical line for clarity. Panel (b) shows the Monte Carlo average over 10,000 replications of $L^*(d)$, $L_{\mu_0}^*(d)$ and $L_{m}^*(d)$. The DGP is given in \ref{['genq1']}-\ref{['genq2']} with $\epsilon_t \sim \textit{NID}(0,1)$, $\omega(L;\varphi_0) = 1$, $d_0 = 0.2$, $\mu_0 = 0$ and $T$ = 64.
  • Figure 2: The approximate and exact intrinsic bias for the ARFIMA(1,$d$,0) model are displayed in panel (a) and (b), respectively. The approximate score bias for the ARFIMA(1,$d$,0) model are displayed in the panels (c), (e) and (g), while the exact score biases are in panels (d), (f) and (h). In all cases, $T = 128$.
  • Figure 3: Density plots of the CSS estimator with unknown level parameter (solid lines), with known level parameter (dashed-dotted lines) as well as of the MCSS estimator (dotted lines) of $d$ (left panels) and $\varphi$ (right panels) in the ARFIMA(1,$d_0$,0) model with $T$ = 32 (upper panels) and $T = 256$ (lower panels), where $d_0 = -0.2$ and $\varphi_0 = 0.5$. The density estimates use a normal kernel.
  • Figure 4: Panel (a) 171 quarterly observations on first differences of log quarterly U.S. real GNP for the time period 1947:2 to 1989:4, as in sowell1992modeling. Panel (b) displays 100 annual observations of the volume of the Nile for the time period 1871 to 1970.
  • Figure 5: The extended Nelson-Plosser data in levels. All of the series are in logs, except for the bond yield.

Theorems & Definitions (80)

  • Theorem 2.1
  • Lemma 2.1
  • Theorem 2.2
  • Corollary 2.1
  • Theorem 2.3
  • Corollary 2.2
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Corollary 2.3
  • ...and 70 more