Table 1.
Nonlinear autoregressive structure of old and modern time series on lynx in the Canadian boreal forest
| No. remaining | Time series | Years | Threshold, θ
|
||||
|---|---|---|---|---|---|---|---|
| NAIC
|
dopt optimal |
Overall θ d’s | Lag-2 | ||||
| d = 1 | d = 2 | ||||||
| L1 | West | 1825–1856 | −0.493 | −0.606 | 2 | 2 | 5.64 (0.56) |
| L2† | West | 1897–1934 | −1.789 | −1.892 | 2 | 2 | 6.56 (0.28) |
| L3* | MacKenzie River | 1821–1934 | −1.416 | −1.472 | 2 | 2 | 7.62 (0.29) |
| L4 | Athabasca Basin | 1821–1891 | −0.367 | −0.198 | 1 | 1 | 4.98 (0.67) |
| L5† | Athabasca Basin | 1897–1934 | −1.455 | −1.356 | 1 | 1 | 7.54 (0.45) |
| L6 | West Central | 1821–1891 | −1.621 | −1.717 | 2 | 2 | 6.18 (0.43) |
| L7† | West Central | 1897–1934 | −1.360 | −1.127 | 1 | 1 | 7.02 (0.36) |
| L8 | Upper Saskatchewan | 1821–1891 | −0.602 | −0.639 | 2 | 2 | 6.19 (0.75) |
| L9 | Winnipeg Basin | 1821–1891 | −1.656 | −1.709 | 2 | 2 | 8.26 (0.36) |
| L10 | North Central | 1821–1891 | −1.231 | −0.911 | 1 | 1 | 5.00 (0.33) |
| L11† | James Bay | 1895–1939 | −2.190 | −2.143 | 1 | 2 (1) | 6.55 (0.41) |
| L12† | Lakes | 1897–1939 | −1.890 | −1.978 | 2 | 2 | 6.59 (0.26) |
| L13† | James Bay and Lakes | 1897–1939 | −2.291 | −2.364 | 2 | 2 | 6.68 (0.32) |
| L14† | Gulf | 1897–1939 | −1.334 | −1.292 | 1 | 1 | 6.22 (0.40) |
| Overall weighted estimates for the Hudson Bay series§ | 6.646 (0.10) | ||||||
| Deduced phase dependency: β1,0 ≤ β2,0 β1,1 ≤ β2,1 β1,2 > β2,2 | |||||||
| Empirical Bayesian estimates for the Hudson Bay series¶ | |||||||
| Deduced phase dependency: β1,1 ≤ β2,1 β1,2 > β2,2 | |||||||
| L15 | British Columbia | 1920–1994 | −1.330 | −1.417 | 2 | 2 | 7.38 (0.26) |
| L16 | Yukon Territory | 1920–1994 | −1.084 | −1.188 | 2 | 2 | 7.25 (0.23) |
| L17 | Northwest Territory | 1920–1994 | −1.462 | −1.460 | 1 | 2 (1) | 7.13 (0.33) |
| L18 | Alberta | 1920–1994 | −1.161 | −1.134 | 1 | 2 (1) | 8.01 (0.43) |
| L19 | Saskatchewan | 1920–1994 | −0.646 | −0.697 | 2 | 2 | 6.51 (0.45) |
| L20 | Manitoba | 1920–1994 | −1.105 | −1.085 | 1 | 2 (1) | 6.39 (0.51) |
| L21 | Ontario | 1920–1994 | −1.960 | −1.928 | 1 | 2 (1) | 6.65 (0.29) |
| L22 | Quebec | 1920–1994 | −2.390 | −2.385 | 1 | 2 (1) | 7.19 (0.27) |
| Overall weighted estimates for the modern series§ | 7.128 (0.11) | ||||||
| Deduced phase dependency: β1,0 ≈ β2,0 β1,1 < β2,1 β1,2 > β2,2 | |||||||
| Empirical Bayesian estimates for the modern series¶ | |||||||
| Deduced phase dependency: β1,1 < β2,1 β1,2 > β2,2 | |||||||
| Grand total weighted estimates for all series§ | 6.858 (0.07) | ||||||
| Deduced phase dependency: β1,0 Η β2,0 β1,1 < β2,1 β1,2 > β2,2 | |||||||
| Empirical Bayesian estimates for all series¶ | |||||||
| Deduced phase dependency: β1,1 < β2,1 β1,2 > β2,2 | |||||||
| The lower regime of the SETAR model, increase phase
|
The upper regime of the SETAR model, decline phase
|
Any trend | ||||
|---|---|---|---|---|---|---|
| β1,0 (±SE) | β1,1 (±SE), direct DD | β1,2 (±SE), delayed DD | β2,0 (±SE) | β2,1 (±SE) direct DD | β2,2 (±SE) delayed DD | |
| 1.30 (1.07) | 1.02 (0.17) | −0.20 (0.30) | 5.83 (2.27) | 1.16 (0.25) | −1.04 (0.36) | No |
| 1.03 (1.09) | 0.91 (0.16) | −0.03 (0.24) | 2.54 (1.37) | 1.71 (0.18) | −1.08 (0.26) | No |
| 1.35 (0.31) | 1.27 (0.06) | −0.43 (0.07) | 2.68 (2.37) | 1.60 (0.13) | −1.01 (0.31) | No |
| 3.10 (2.30) | 0.53 (0.30) | −0.05 (0.44) | 3.52 (0.64) | 1.33 (0.10) | −0.86 (0.12) | No |
| 4.10 (1.08) | 1.36 (0.16) | −0.94 (0.18) | 5.39 (1.39) | 1.34 (0.16) | −0.99 (0.22) | No |
| 1.13 (0.62) | 1.28 (0.08) | −0.35 (0.13) | 1.71 (0.59) | 1.52 (0.08) | −0.81 (0.10) | No |
| 4.59 (1.73) | 0.80 (0.25) | −0.51 (0.26) | 3.97 (2.00) | 1.31 (0.19) | −0.83 (0.28) | No |
| −0.05 (1.36) | 1.08 (0.17) | 0.10 (0.92) | 2.90 (0.81) | 1.40 (0.10) | −0.81 (0.13) | No |
| 2.42 (0.80) | 1.37 (0.09) | −0.64 (0.13) | 1.91 (1.53) | 1.42 (0.13) | −0.67 (0.19) | No |
| 2.98 (1.31) | 0.76 (0.19) | −0.31 (0.25) | 0.48 (0.83) | 1.44 (0.11) | −0.59 (0.15) | No |
| 1.65 (0.46) | 1.45 (0.09) | −0.70 (0.11) | 1.60 (1.88) | 1.44 (0.18) | −0.73 (0.30) | No |
| 2.79 (0.95) | 1.29 (0.16) | −0.75 (0.20) | 4.79 (1.86) | 1.33 (0.19) | −0.99 (0.28) | No |
| 3.30 (1.02) | 1.30 (0.12) | −0.80 (0.18) | 3.68 (1.06) | 1.56 (0.14) | −1.05 (0.17) | No |
| 0.97 (1.07) | 1.05 (0.18) | −0.23 (0.23) | 2.09 (2.44) | 0.92 (0.24) | −0.31 (0.40) | No |
| 1.70 (0.19) | 1.24 (0.03) | −0.50 (0.04) | 2.63 (0.29) | 1.43 (0.04) | −0.82 (0.05) | |
| 1.25 (0.11) | −0.54 (0.13) | 1.40 (0.05) | −0.79 (0.07) | |||
| −1.15 (2.21) | 0.17 (0.17) | 0.76 (0.29) | 1.81 (1.28) | 0.96 (0.15) | −0.20 (0.20) | No |
| 3.20 (1.23) | 0.79 (0.16) | −0.26 (0.21) | 2.19 (1.82) | 1.25 (0.13) | −0.58 (0.26) | No |
| 6.32 (3.58) | 0.53 (0.28) | −0.42 (0.43) | 4.29 (0.98) | 1.07 (0.12) | −0.63 (0.15) | No |
| 1.97 (1.10) | 0.88 (0.16) | −0.14 (0.18) | 3.52 (1.18) | 1.45 (0.13) | −0.86 (0.17) | No |
| 4.90 (1.66) | 0.27 (0.22) | −0.11 (0.23) | 1.75 (0.63) | 1.27 (0.13) | −0.51 (0.14) | No |
| 2.86 (1.32) | 0.76 (0.22) | −0.27 (0.24) | 2.09 (0.60) | 1.29 (0.12) | −0.58 (0.13) | No |
| 5.76 (3.35) | 0.51 (0.28) | −0.40 (0.41) | 2.88 (0.67) | 1.26 (0.11) | −0.65 (0.13) | (<0.10) |
| 2.67 (1.33) | 1.31 (0.16) | −0.70 (0.22) | 4.13 (0.90) | 1.30 (0.11) | −0.83 (0.14) | No |
| 2.80 (0.54) | 0.75 (0.07) | −0.20 (0.09) | 2.66 (0.30) | 1.24 (0.04) | −0.63 (0.05) | |
| 0.92 (0.17) | −0.15 (0.12) | 1.21 (0.13) | −0.47 (0.20) | |||
| 1.83 (0.18) | 1.15 (0.03) | −0.45 (0.04) | 2.64 (0.20) | 1.36 (0.03) | −0.73 (0.04) | |
| 1.10 (0.20) | −0.37 (0.22) | 1.32 (0.12) | −0.66 (0.17) | |||
Assuming a SETAR(2;2,2) model, the NAIC [NAIC being AIC = −2ln(max likelihood) + 2(number of parameters) normalized by the effective number of observations] values for d = 1 and d = 2 are given together with the optimal d value, dopt, defined as the one minimizing the NAIC over d = 1 and d = 2; in cases that the NAIC values for d = 1 and d = 2 are insignificantly different {defined by [(NAIC(d−)−NAIC(dopt)]/[−NAIC(dopt)] < 0.025}, where d− is the nonoptimal d, both 1 and 2 are listed; d = 2 is given in bold because this is the overall most appropriate delay. The estimated parameters in the SETAR model (Eq. 1) for the lynx time series from Canada are provided by Elton and Nicholson (1) [L1-L14] and the time series provided by Dominion Bureau of Statistics and Statistics Canada (36, 37) [L15-L22]. Analyses are based on the original and not detrended data, for which the thresholds are estimated on the basis of the NAIC criterion; the same conclusions emerge if detrended data are analyzed. Detrending was done in S-plus by subtracting a fitted cubic B-spline with 4 degrees of freedom (38). The optimal threshold, θ, assuming a lag (d) equal to 2, was determined by NAIC (Ref. 39; p. 379); the threshold estimate is given together with the estimated bootstrap SE (40, 41). The column “Any trend” summarizes the results of testing the null hypothesis of a SETAR(2;2,2) model against the alternative of a “SETAR(2;2,2) + linear time trend” model. “No” indicates a rejection of the alternative at 5% level and hence suggests the adequacy of a SETAR model. The test is implemented via the method of Lagrange multiplier, also known as the score method (42). The overall weighted estimates were calculated as weighted means, Σ(μtwi)/Σwi, where μi are the estimated parameters for series i and wi = 1/(SEi)2. The overall SE is given as (1/Σwi)1/2. The empirical Bayes estimation (43) is done via the EM-algorithm (44). All series first are normalized (linearly) so that the 30 (70) percentiles become 0 (1). Only the mean lag-1 and lag-2 coefficient estimates are given in the table because the other parameters are not invariant under the scale change.
SE, standard error; NAIC, normalized kaike information criterion; DD, density dependent.
This series was analyzed by Tong (39).
Series has been interpolated for the missing observation in year 1914.
This combined series was studied by Stenseth et al. (28, 29) because this most closely corresponded to the snowshoe hare series they studied; this combined series is included here for comparative reasons but is excluded from the both sets of pooled estimates.
The weighted estimates are computed under the framework that the SETAR coefficients are the same for the all of the series in a particular panel of lynx data. The numbers in parentheses are the standard errors of the weighted estimates.
The empirical Bayesian estimates are computed based on a random coefficient model that for each series the SETAR coefficients are drawn from a super-population. The numbers in parentheses are the corresponding (between-region) standard deviations of the super-population (see main text).