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Nonfractional Memory: Filtering, Antipersistence, and Forecasting

J. Eduardo Vera-Valdés

Abstract

The fractional difference operator remains to be the most popular mechanism to generate long memory due to the existence of efficient algorithms for their simulation and forecasting. Nonetheless, there is no theoretical argument linking the fractional difference operator with the presence of long memory in real data. In this regard, one of the most predominant theoretical explanations for the presence of long memory is cross-sectional aggregation of persistent micro units. Yet, the type of processes obtained by cross-sectional aggregation differs from the one due to fractional differencing. Thus, this paper develops fast algorithms to generate and forecast long memory by cross-sectional aggregation. Moreover, it is shown that the antipersistent phenomenon that arises for negative degrees of memory in the fractional difference literature is not present for cross-sectionally aggregated processes. Pointedly, while the autocorrelations for the fractional difference operator are negative for negative degrees of memory by construction, this restriction does not apply to the cross-sectional aggregated scheme. We show that this has implications for long memory tests in the frequency domain, which will be misspecified for cross-sectionally aggregated processes with negative degrees of memory. Finally, we assess the forecast performance of high-order $AR$ and $ARFIMA$ models when the long memory series are generated by cross-sectional aggregation. Our results are of interest to practitioners developing forecasts of long memory variables like inflation, volatility, and climate data, where aggregation may be the source of long memory.

Nonfractional Memory: Filtering, Antipersistence, and Forecasting

Abstract

The fractional difference operator remains to be the most popular mechanism to generate long memory due to the existence of efficient algorithms for their simulation and forecasting. Nonetheless, there is no theoretical argument linking the fractional difference operator with the presence of long memory in real data. In this regard, one of the most predominant theoretical explanations for the presence of long memory is cross-sectional aggregation of persistent micro units. Yet, the type of processes obtained by cross-sectional aggregation differs from the one due to fractional differencing. Thus, this paper develops fast algorithms to generate and forecast long memory by cross-sectional aggregation. Moreover, it is shown that the antipersistent phenomenon that arises for negative degrees of memory in the fractional difference literature is not present for cross-sectionally aggregated processes. Pointedly, while the autocorrelations for the fractional difference operator are negative for negative degrees of memory by construction, this restriction does not apply to the cross-sectional aggregated scheme. We show that this has implications for long memory tests in the frequency domain, which will be misspecified for cross-sectionally aggregated processes with negative degrees of memory. Finally, we assess the forecast performance of high-order and models when the long memory series are generated by cross-sectional aggregation. Our results are of interest to practitioners developing forecasts of long memory variables like inflation, volatility, and climate data, where aggregation may be the source of long memory.

Paper Structure

This paper contains 13 sections, 5 theorems, 29 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The Fractional Difference Operator
  3. Cross-Sectional Aggregation
  4. Nonfractional Memory Generation
  5. Nonfractional Memory and Fractional Difference Operator
  6. Nonfractional Memory and Antipersistent Processes
  7. Nonfractional Memory Forecasting
  8. Minimum Square Error Forecasts
  9. Forecasts With AR(p) Models
  10. Forecasts With Fractional Models
  11. Conclusions
  12. Proofs
  13. Code Implementation

Key Result

Theorem 1

Let $x_t\sim CSA(a,b)$ defined as in (eq:csa_def), then $x_t$ can be computed as the first $T$ elements of the $(2T-1)\times 1$ vector where $\bar{F}$ is the discrete Fourier transform, $T^{-1}F$ is the inverse transform, and '$\odot$' denotes multiplication element by element. Furthermore, $\tilde{z}$ is a $(2T-1)\times 1$ vector given by $\tilde{z}:=[z_0, z_1, \cdots, z_{T-1},$$0, \cdots, 0]$ f

Figures (5)

  • Figure 1: Autocorrelation function for an $CSA(a,b)$ processes for different values of '$a$' while having the same asymptotic behaviour.
  • Figure 2: Filtered series and autocorrelation function for a fractional differenced process $I(0.2)$, and a cross-sectional aggregated one $CSA(0.12,1.6)$.
  • Figure 3: Autocorrelation functions for a fractional differenced process, $I(-0.2)$, and a cross-sectional aggregated one, $CSA(0.09,2,4)$. The right plot shows a close-up for lags 70 to 110.
  • Figure 4: Mean periodogram of a fractional differenced process, $I(d)$, [top], and a cross-sectional aggregated process, $CSA(0.2,2(1-d))$, [bottom]. A sample size of $T=10,000$ was used, with 10,000 replications, and $d=0.4,-0.4$.
  • Figure 5: Estimated autoregressive parameter and forecast error variance of fitted $AR(1)$ on $CSA(a,b)$ processes relative to the optimum forecasts while varying '$a$', for different degrees of memory.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5