An empirical study on the parsimony and descriptive power of TARMA models

Abstract

In linear time series analysis, the incorporation of the moving-average term in autoregressive models yields parsimony while retaining flexibility; in particular, the first order autoregressive moving-average model, ARMA(1,1) is notable since it retains a good approximating capability with just two parameters. In the same spirit, we assess empirically whether a similar result holds for threshold processes. First, we show that the first order threshold autoregressive moving-average process, TARMA(1,1) exhibits complex, high-dimensional, behaviour with parsimony, by comparing it with threshold autoregressive processes, TAR(p), with possibly large autoregressive order p. Second, we study the descriptive power of the TARMA(1,1) model with respect to the class of autoregressive models, seen as universal approximators: in several situations, the TARMA(1,1) model outperforms AR(p) models even when p is large. Lastly, we analyze two real world data sets: the sunspot number and the male US unemployment rate time series. In both cases, we show that TARMA models provide a better fit with respect to the best TAR models proposed in literature.

Appendix: Supplementary results

1.1 The parsimony of the TARMA(1,1) model

See Tables 15, 16, 17 and 18.

Table 15 Distribution of the autoregressive order p of the TAR(p), selected through the AIC, for $n=200$. Each row corresponds to a different parameter selection for the TARMA(1,1) DGP

Table 16 Distribution of the autoregressive order p of the TAR(p), selected through the AIC, for $n=500$. Each row corresponds to a different parameter selection for the TARMA(1,1) DGP

Table 17 Distribution of the autoregressive order p of the TAR(p), selected through the BIC, for $n=200$. Each row corresponds to a different parameter selection for the TARMA(1,1) DGP

Table 18 Distribution of the autoregressive order p of the TAR(p), selected through the BIC, for $n=500$. Each row corresponds to a different parameter selection for the TARMA(1,1) DGP

1.2 The descriptive power of TARMA models

See Tables 19 and 20 and Figs.10, 11, 12 and 13

Fig. 10

Boxplots of the ratio of information criteria between the TARMA and the TAR(1) model (upper panel) and between the TARMA and the TAR(3) model (lower panel) versus sample size n, for model ARIMA(1,1,1)

Fig. 11

Boxplots of the ratio of information criteria between the TARMA and the TAR(1) model (upper panel) and between the TARMA and the TAR(3) model (lower panel) versus sample size n, for model TAR(3)

Fig. 12

Boxplots of the ratio of information criteria between the TARMA and the TAR(1) model (upper panel) and between the TARMA and the TAR(3) model (lower panel) versus sample size n, for model NLMA.1

Fig. 13

Boxplots of the ratio of information criteria between the TARMA and the TAR(1) model (upper panel) and between the TARMA and the TAR(3) model (lower panel) versus sample size n, for model NLMA.2

Table 19 Percentages of selected $p_\mathrm{{AIC}}$ for different DGPs and $n=200,500$

Table 20 Percentages of selected $p_\mathrm{{BIC}}$ for different DGPs and $n=200,500$

1.3 Sunspots

In Table 21 we extend the analysis reported in Table 13 by varying the delay $d=1,3,5$. We chose such values for d in accordance with the previous literature (see, for instance, Tong and Lim 1980). Since we use different values for d, the effective sample size changes so that information criteria are not comparable. To cope with this issue, we use the normalized AIC and BIC (see Tong 1990, p. 379.) We denote the normalized information criteria by NAIC and NBIC. The best fits are the TARMA(22,1)c and the TAR(22)b with $d=1$.

Table 21 Best fits for the sunspot index: lags, d, NAIC and NBIC. The lags column indicate the subset of lagged variables that form the TAR component in both regimes; d is the delay parameter