Fitting a linear multivariate model with alpha set at 0.05 (Model D

Now we fit a model to the dataset where alpha is set at 0.05 to conform to the standard practice in economics and social sciences. This model will be one of the models providing “best estimates”. The SAS command used to generate the output is the same as the one used to fit Model C except that the “alpha = 0.05” option is included in the “proc logistic” statement and the ‘entry and stay’ conditions are removed. Tables 6.15a to 6.15e summarize the output of the logit regression procedure.

The AIC, SC, and -2Log L indicate that the model with covariates fits better than the ‘empty model’ (model with intercept only). The Residual Chi-Square test rejects the null hypothesis that the empty model is adequate. The fitted model with covariates has a Chi-Square of 11.1531 with 9 degrees of freedom and a p-value of 0.2654 indicating that the model fits the data well. The model R2 of 0.3278 indicates that 32.78% of the total variations in the dependent variable may be explained in terms of explanatory variables included in the model. The Hosmer and Lemeshow Goodness-of-Fit Test with a Chi-Square of 2.2024, 8 degrees of freedom and a p-value of 0.9742 is a strong

indication that the fitted model is adequate. The predictive capacity of the fitted model is good at 79%.

Table 6.15a Model Fit Statistics for logit Model D

Criterion Intercept Only Intercept and Covariates

AIC 319.940 265.014

SC 323.383 285.669

-2 Log L 317.940 253.014

Table 6.15b Maximum likelihood estimates and model fit statistics for logit Model D Parameter DF Estimate Standard

Error Wald Chi- Square Pr > ChiSq Intercept 1 -3.0834 1.1716 6.9269 0.0085 Score 1 0.4847 0.2069 5.4854 0.0192 Supports 1 2.6462 1.0661 6.1609 0.0131 Seed 1 -0.0147 0.00342 18.4528 <.0001 Distance 1 -0.0153 0.00686 4.9873 0.0255 Activity2 1 0.0895 0.0282 10.0498 0.0015

Testing Global Null Hypothesis: Beta = 0

Test Chi-Square DF Pr > ChiSq LR 64.9257 5 <.0001 Score 51.3125 5 <.0001 Wald 38.0346 5 <.0001 Residual Chi-Square 11.1531 9 0.2654 Hosmer and Lemeshow Goodness-of-Fit Test 2.2024 8 0.9742 R-Square 0.2450 Max-rescaled R-Square 0.3278 Table 6.15c Odds Ratio Estimates for logit Model D

Effect Point Estimate 95% Wald Confidence Interval Lower CL Upper CL Score 1.624 1.082 2.436 Supports 14.100 1.745 113.941 Seed 0.985 0.979 0.992 Distance 0.985 0.972 0.998 Activity2 1.094 1.035 1.156

Table 6.15d Association of predicted probabilities and observed responses for logit Model D

Percent Concordant 79.0 Somers' D 0.582 Percent Discordant 20.8 Gamma 0.583 Percent Tied 0.2 Tau-a 0.290 Pairs 13208 c 0.791 Table 6.15e Partition for the Hosmer and Lemeshow Test for logit Model D YDC =1 YDC = 2

Group Total Observed Expected Observed Expected

1 23 1 0.61 22 22.39 2 23 3 3.11 20 19.89 3 23 5 5.65 18 17.35 4 23 8 7.98 15 15.02 5 23 9 10.43 14 12.57 6 23 12 12.00 11 11.00 7 23 15 13.46 8 9.54 8 23 16 14.77 7 8.23 9 23 17 16.42 6 6.58 10 24 18 19.59 6 4.41

The intercept and coefficients on Support, Seed, and Activity2 are significant at the .01 level, while distance and Score are significant at the .05 level. The variables in the model have the expected signs.

6.2.4.1 Estimating WTP from the Fitted Logit Model D

Applying equations (7b), (9), and (10) to the fitted model yields estimates of truncated mean, mean (or median), and mean WTP of NZ$69.26, NZ$49.53, and NZ$76.33 respectively. Equation (10) produces a higher estimate as expected. Models C and D yield similar estimates. The ‘quasi-confidence’ interval for WTP per household per year is NZ$34.12 to NZ$90.42. The parameters of the fitted logit model and their corresponding transformed coefficients were used to construct a WTP function and a WTP probability function as given below;

WTPi= -209.75 + 32.97*Scorei + 180.01*Supportsi - 1.04*Distancei

+ 6.09*Activity2i (15a)

A plot of predicted probabilities against the corresponding dollar Seed values generates a WTP probability curve as depicted in Figure 6.7. When the suggested bid amount is zero, we expect the probability of a “yes” response to be equal to or close to 1, assuming that the improvement is not a disutility to some respondents. The predicted probability of a “yes” response, derived from the fitted model, when the bid set at zero is approximately 0.674 indicates that some respondents are indifferent to the wetland improvement programme even when the programme costs them nothing. Respondents who are indifferent when the bid amount is zero include those who do not enjoy direct and indirect use value of wetlands in general; those who reside too far from Pekapeka Swamp and are unlikely to benefit from it in future; and those who have no concern for environmental protection.

The predicted probability of a “yes” response of 0.674 may also suggest that the WTP distribution in the population from which our sample was drawn from includes negative WTP values i.e. some respondents require compensation if the programme is implemented. However the negative values of WTP that may be estimated from the fitted model should not to be used as estimates of WTA as they differ from true WTA (because none of the respondents were asked a WTA question) and would grossly underestimate it.

An estimate of the sample median WTP may be estimated from the graph by reading off the Seed amount corresponding to Pr (yes) = 0.5. The Seed amount corresponding to Pr (yes) = 0.5 is approximately $50.00 and is close to the mean (or median) of $49.53 estimated using equation (9). The slight difference of about $0.47 may be attributed to the scale on the x-axis, which lacks sensitivity to small changes in the Seed value. The shape of the graph of the WTP probability function (Figure 6.7) when extended to envelop negative WTP values suggests that equations (7) and (10) overstate the mean estimates by excluding the checked area above the curve.