2.4 Results
2.4.2 Additional specification and diagnostic checks
The parallel regression assumption, which states that all coefficients (excluding the cut-off points) are equal across all J categories of the ordered dependent variable, is revisited in this subsection as the likelihood ratio test of the parallel regression assumption indicates it is violated (Chi2=111.12, p=0.00). However, based on the
results that follow, the parallel assumption does not hold for only a small number of variables. This subsection implements two models to check whether the preceding findings hold while relaxing the parallel assumption: the partial parallel regression model and heterogeneous choice model (see, e.g., Williams, 2006; Williams, 2010). There are several models that can help address a violation of the parallel assumption. The multinomial probit, which would completely relax this assumption (i.e. allows all coefficients to vary across J categories), is one alternative. However, it is likely to be 1) unreliable as one would be estimating over 100 coefficients with only 784 observations (i.e. fewer than 10 per parameter), and 2) inefficient due to dispensing of the information contained in a naturally ordered dependent variable. A set of generalised ordered models can help relax the parallel assumption, without throwing the baby out of the bathwater.
The partial parallel regression model acknowledges that a specific set of coefficients may vary over J categories. It is possible to test whether the proportional assumption holds by testing the null hypothesis that the deviation g from proportionality for each variable equals zero (Peterson and Harell, 1990). If the null hypothesis cannot be rejected for all variables (i.e. g=0), then the model reverts to a standard ordered regression. The fully nested model in Section 2.3.3 can be rewritten as follows:
P(Di= j) = Φ(αj- Xi'" - Vi'#$) - Φ(αj-1- Xi'" - Vi'#$)
Where V is a vector containing the variables that do not satisfy the parallel assumption and the vector of coefficients g measure the deviation from proportionality from the baseline group for each j-1 categories. (All controls are contained in the vector X for notational simplicity.)
An (empirical) backward stepwise approach is followed to identify the variables that may not satisfy the parallel assumption34. The variables found to be problematic within a partial parallel framework are: INATTENTIVE B and UNSURE BILLS/INCOME35. Hence, Table 8 (Column 8) presents the estimate of the deviation
from parallel assumption g for these variables. The implied discount rate only falls by around 3 percentage points compared to the results presented in Column 1 (Table 8). An alternative approach investigates whether the variation in coefficients is just a product of heterogeneity in the error variance, i.e. the heterogeneous choice model (Williams, 2006; Williams, 2010). To see how heterogeneity can bias the estimated coefficients one can rescale the coefficients by a factor of s which represents the ‘adjustment [for the] error variance’ (Williams, 2010: 3), as follows:
P(Di= j) = Φ αj - Xi 'δ σi - Φ αj-1- Xi'δ σi
34 Stata’s ‘stepwise’ command (setting the threshold a=0.01), also identifies the same variables as a
backward stepwise approach implemented by hand. However, there is very little difference in the results when setting the threshold to a=0.10, hence the results presented use a 10% threshold.
Given homogeneity the scaling parameter would simply equal 1, and the model would once again return to the original ordered regression. If the error is found to increase or decrease in X (or any other variable) the original coefficients will be biased across groups – potentially giving the impression that the deviation from the parallel assumption is due to actual differences across groups when the root cause may instead be heterogeneity.
Using a backward stepwise approach, the error variance appears to adjust with: PBK-
H, UNSURE BILLS/INCOME, and KNOWS DH x INTERFACE36. Table 8 (Column
7) presents the coefficient j estimates (for these variables) implied by the error variance scaling parameter s, since s=Z’j, whereby Z represents the vector of variables that are significantly related to the variance of the error term. The implied discount rate increases by around 3 percentage points compared to the fully nested model (Column 1, Table 8).
The implied discount rates presented in Table 8 (Columns 7 and 8), 0.33 and 0.38 respectively, are qualitatively very similar to those generated by the ordered probit model (Column 1). While, the LR test, AIC, BIC, and Pseudo-R2, suggest very little
difference in terms of the fit between the heterogeneous choice and partial models (Columns 7 and 8); the BIC supports the ordered probit and heterogeneous model over the partial regression – though the latter marginally outperforms all other models in terms of its predicted probabilities (Appendix A5, Table A.5). Nonetheless, the BIC favours the ordered probit model (1) over the heterogeneous model (7) and
36 The p-values of the ‘stepwise’ procedure are as follows: PBK-H (0.00), UNSURE BILLS/INCOME
partial parallel model (8), thereby an out-and-out rejection of the parallel assumption is inconclusive.
In light of the preceding results, a series of diagnostic tests/alternative specifications are presented for the classic model (Column 4) using the heterogeneous choice model (all of which are described in Table A.2 and presented in Table A.3-A.4 in Appendix A4). In doing so, one can seek to uncover whether the discount rate remains stable and significant (at least at the 10% level), thereby shedding light on whether the change in the discount rate and its significance is due to either the inclusion of the behavioural variables or simply to variation (or uncertainty) in the underlying empirical framework. The spread of the average implied discount rate is displayed in Figure 1.
Figure 1: Spread of the implied discount rate
Fr
eq
ue
nc
y
Implied Discount Rate Mean=0.394 S.D.=0.052 M=17
Bearing in mind the limited number of models (M=17), the results show that the mean of the implied discount rate is around 0.39 with a standard deviation of 0.052. Therefore, homeowners appear indifferent between an upfront cost of around £0.39
0 2 4 6 8 F re q u e n cy .3 .35 .4 .45 .5
and a £1 change in discounted annual bill37. One concedes however that the
diagnostic tests do not support a single model and thereby one cannot identify a single implied discount rate, instead a reassuringly small window is found in which this rate falls.
The findings suggest that a range of models of consumer behaviour with regard to energy efficiency investments exist. Whereby, using a classical approach, households’ decisions fall in line with an aggregate discount rate that is higher than what one would expect in a neo-classical framework. However, according to the findings presented in Table 8 inattention explains around 10 percentage points of the discount rate before controlling for heuristic decision-making and 4 percentage points after. Therefore, the results are indicative of an economy containing households bounded in their rationality, and who choose to simplify the decision-making process, by relying on mental shortcuts or a subset of information, since the cost of either acquiring or internalising the extra information is larger than the benefits of using the full set of information or exceeds the consumer’s cognitive capabilities within a given timeframe (Golove and Eto, 1996; Gigerenzer and Gaissmaier, 2011).