Vtd Yl

where transfer function operations are performed to identify the lag polynomials v(B) and w(B) of degree Zj and rrij (Box & Jenkins, 1970; Makridakis & Wheelwright, 1978).

The AR and MA structures were chosen through a lengthy trial and error process. It was found that the most efficient model demonstrated an autoregressive structure of p = 1 and a moving average operator of q = 8, for d = 1 (the first-differenced data). The following explanatory variables were then used to account for the remaining variation in yt (in various combinations thereof) using the MARMA (p = 1, q = 8, d= 1) model:

Xlt = the dummy variable D(t)\ X2t — average sea surface temperature ASST(i); and X3t — an additional dummy variable DD(t) denoting the El Nino years of 1965, 1969, 1972, 1977, 1982, and 1986, where DD(t) = 1; and DD(t) = 0 for the remaining years.

These additional explanatory variables are supposed to capture the impact of changing environmental conditions on the availability of fish stock. Nonlinear Gaussian estimation was performed using the econometrics package SORITEC (version 6.3, 1989), and the statistically relevant findings and estimated parameters are listed in Table 15.1. An illustration of the actual and simulated

Table 15.1

Output for MARMA model of Peruvian and Chilean fishing industries for the period 1951-86

Table 15.1

Output for MARMA model of Peruvian and Chilean fishing industries for the period 1951-86

Coefficient

Estimated

Standard

t-

description

coefficient

error

statistic

X\ (Dummy)

0i

-0.0782698

0.016827

-4.6512*

X2 (ASST)

02

-0.0096424

0.004186

-2.3032*

AR-term(-l)

4>i

0.707452

0.174205

4.0610*

MA-term(-l)

©1

0.957460

0.197138

4.8568*

MA-term(-2)

©2

-0.557181

0.259636

-2.1460*

MA-term(-3)

©3

0.354013

0.288278

1.22803

MA-term(-4)

©4

-0.428059

0.199701

-2.14350*

MA-term(-5)

©5

0.891684

0.187595

4.75324*

MA-term(-6)

©6

-0.604302

0.259006

-2.33315*

MA-term(-7)

©7

-0.022545

0.284620

-0.07921

MA-term(-8)

©8

0.418544

0.202160

2.07036*

*Significant at the 95% confidence level (|i| > 1.714)

*Significant at the 95% confidence level (|i| > 1.714)

Summary Statistics:

Sum of Squared Residuals = 0.006054 Variance of Residuals = 0.0002632 Durbin-Watson Statistic = 1.9234 Box-Pierce Q Statistics (30 lags) = 8.73 R-Squared = 0.9488

i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 1955 I960 1965 1970 1975 1980 1985 YEAR

Fig. 15.8 First-differenced time series, actual and simulated fish catch.

i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 1955 I960 1965 1970 1975 1980 1985 YEAR

Fig. 15.8 First-differenced time series, actual and simulated fish catch.

first-differenced time series appear in Fig. 15.8, demonstrating the close fit produced by this model.

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