Long Memory deals with the notion that certain series have autocorrelation functions that decay slower than what standard models can account for. The autocorrelation function for a long memory process shows hyperbolic decay, which translates into shocks having non-zero effects even after much time has passed. As such, its presence has repercussions for inference and prediction.

Long Memory has been detected in several time series. To name a few, in Economics, in inflation and GDP; in Finance, in volatility measures and electricity prices; in Climate Econometrics, in temperature and river flows.

As an example, the figure presents the medians of the monthly temperature series of the Northern Hemisphere and its autocorrelation function. The series is presented as temperature anomalies relative to 1961-1990. As shown by its autocorrelation function, the series presents long memory. That is, the autocorrelation function is still significant after 100 lags. In particular, autoregressive processes are not capable of capturing the full dynamics.

Cross-Sectional Aggregation

One of the main theoretical arguments behind the presence of long memory in real data is cross-sectional aggregation.

The core idea relies in the fact that cross-sectional aggregation of autoregressive processes can result in long memory properties in the aggregate.

The argument is relevant given that several time series are constructed by aggregation of individual series. A prime example is inflation or regional temperature averages (see above).

In my research, I have analysed several aspects of the cross-sectional argument for long memory:

• Long Memory, Fractional Integration, and Cross-Sectional Aggregation shows that processes generated by cross-sectional aggregation satisfy all long memory properties discussed in the literature. Moreover, it shows that the long memory generated by cross-sectional is not a member of kind of processes generated by the fractional difference operator.

• On Long Memory Origins and Forecast Horizons shows that the fractional difference operator obtains good forecasting performance when working with long memory generated by non-fractional methods.

• Nonfractional Long-Range Dependence: Long Memory, Antipersistence, and Aggregation uses aggregation as inspiration to develop a model to generate long-range dependent processes that does not depend on the fractional difference operator. Our model thus circumvents Granger’s concern that the fractional difference operator is in the empty box of models that do not arise in the actual economy. Moreover, we show that the generated processes are less brittle than fractionally integrated processes.

Spurious Regressions

The presence of long memory can have perverse effects if not properly addressed in the modelling scheme. I have worked on some of these effects and how to remediate them.

• The VIX, the Variance Premium, and Expected Returns shows that inconsistency arises from unbalanced regressions due to the presence of long memory. The paper proposes a 2SLS estimator that produces consistent estimates.

• Spurious Multivariate Regressions Under Fractionally Integrated Processes shows that the asymptotic behavior from the most persistent regressor spreads to correlated regressors. Thus, the risk of uncovering spurious results increases as more regressors are included.

Applications

• The Persistence of Financial Volatility After COVID-19 studies the long memory properties of several international volatility measures before and after the emergence of the pandemic. The paper shows that volatility become more persistent after COVID-19, signalling the long-term effects of the pandemic on the finance sector.

• Temperature Anomalies, Long Memory, and Aggregation studies the long memory properties of individual grid temperatures and compares them against global and regional aggregates. The analysis shows that aggregation may be exacerbating the long memory properties of temperature data.

• Long-Lasting Economic Effects of Pandemics: Evidence from the United Kingdom analyses the long-term economic effects of previous pandemics. The paper shows that shocks to growth and unemployment during and right after the pandemic have long memory properties. Thus, V-shaped recoveries have not been the norm in previous pandemics.