Investigating the Long Memory Process in Daily High School Attendance Data

Abstract

While high school attendance is a well-known mediator in the relationship between educational attainment, background variables such as poverty and parental support on the one hand and instructional and school policy variables on the other, the study of high school attendance as a variable of interest in its own right is extremely rare in education. At most, it is reported in aggregated form without an estimation of its variability over time. The analysis presented in this chapter demonstrates how nonlinear time series analysis, autoregressive fractionally integrated moving average (ARFIMA) methods in particular, can be used to analyze high school daily attendance rates over a long time period (7 years) to obtain a fine-grained assessment of the time dependency of attendance behavior. The chapter first discusses through simulated examples how error dependencies can play out over a longer time spectrum (e.g., short-term autoregression, pink noise, Brownian motion), and then presents an analysis of real daily high school attendance data to illustrate these dependencies. The analyses show evidence of self-organized criticality (pink noise) in some but not all schools analyzed, a finding that indicates that attendance rates are more stable and predictable in some schools than in others.

Appendices

Appendix 1: Short-Range Estimation Using ARIMA

The general model AR can be stated as

$$ {Y}_t = {\phi}_1{Y}_{t-1} + {\phi}_2{Y}_{t-2}+\cdots + {\phi}_p{Y}_{t-p} + {e}_t $$

This model estimates Y _t using p lags. The parameter ϕ estimates the influence of past observations on the series at each given lag.

The MA model estimates Y _t in terms of accumulated error disturbances, also called innovations. Using q lags, this estimation can be written as follows:

$$ {Y}_t = {e}_t - {\theta}_1{e}_{t-1} - {\theta}_2{e}_{t-2}-\cdots - {\theta}_q{e}_{t-q} $$

In this equation θ estimates the impact of each innovation on the series.

AR and MA processes can be captured in a single predictive model. For purposes of clarity, we describe a predictive model that uses one lag only, i.e., p = 1 and q = 1:

$$ {Y}_t={\phi}_1{Y}_{t-1}+{e}_t-{\theta}_1{e}_{t-1} $$

A special case is the seasonal ARMA process, which estimates the dependencies in terms of days of the week, months in a year, etc. The analysis presented here focuses on the regularities as a cyclical weekly pattern with 5 days in the school week. The model used to address this question can be formally written as

$$ {Y}_t={\phi}_1{Y}_{t-5}+{e}_t-{\theta}_1{e}_{t-5} $$

The autocorrelation function (ACF) at lag k is defined as

$$ {r}_k = {\displaystyle \sum_{t=k+1}^n}\left({Y}_t - \overline{Y}\right)\left({Y}_{t-k} - \overline{Y}\right)\ /\ {\displaystyle \sum_{t=1}^n}\left({Y}_t-\overline{Y}\right)\kern1em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em k=1,2,\dots $$

Appendix 2: Long-Range Estimation Using ARFIMA

Some mathematical reorganization of the terms in the ARIMA model as stated in Appendix 1 is required to describe what the estimation of the long-range influences adds to the models that assess the short-range effects on attendance trajectories.

It is often conventional in time series notation to express ARMA processes in terms of the so-called lag operator, or backshift operator, which is defined as

$$ B{Y}_t = {Y}_{t-1} $$

In plain English, the backshift operator B shifts observations back one time unit to construct a new series. The next lag over can be written as BBY _t  = Y _t−2, or

$$ {B}^2{Y}_t = {Y}_{t-2} $$

In terms of this operator, the ARIMA process described above is often written as

$$ \left(1+{\varphi}_1B+{\varphi}_2{B}^2+\cdots +{\varphi}_p{B}^p\right){Y}_t = \left(1+{\theta}_1B+{\theta}_2{B}^2+\cdots +{\theta}_q{B}^q\right){e}_t $$

The left side of the equation represents the autoregression (AR) component; the moving average (MA) component is on the right. The mathematical derivation of this formulation, called the characteristic equation, from the equations above can be found in Box and Jenkins (1970), Cryer and Chan (2008), and many other standard time series texts. It is assumed in this model that remaining error is randomly distributed, i.e.,

$$ {e}_t\ \left(t=1,\ 2, \dots \right)\kern0.5em \sim N\left(0,\ {\sigma}^2\right)\ \mathrm{I}\mathrm{I}\mathrm{D}. $$

The ARFIMA model separates long-term dependencies from the short-term ones by parameterizing d as a differencing estimate:

$$ \left(1+{\varphi}_1B+{\varphi}_2{B}^2+\cdots +{\varphi}_p{B}^p\right){\left(1\hbox{-} B\right)}^d{Y}_t=\left(1+{\theta}_1B+{\theta}_2{B}^2+\cdots +{\theta}_q{B}^q\right){e}_t $$

It is assumed here that the trajectory is stationary and that −0.5 < d < 0.5 (Beran, 1994; Sowell, 1992; Stadnitski, 2012b).