Table 2.

Simulation Results for Ordered Categorical Outcome with Count Endogenous Variable Average Absolute % Bias

Average Effects	Based on 1000 replicates of size n=10,000
	True Model (%bias)	Naïve Model (%bias)	2SPS Model (%bias)	2SRI Model (%bias)
P(y1(xe = ln(4)) = 1) − P(y1(xe = ln(2)) = 1)	0%	20%	28%	4%
P(y2(xe = ln(4)) = 1) − P(y2(xe = ln(2)) = 1)	1%	67%	29%	1%
P(y3(xe = ln(4)) = 1) − P(y3(xe = ln(2)) = 1)	15%	367%	44%	19%
P(y4(xe = ln(4)) = 1) − P(y4(xe = ln(2)) = 1)	3%	191%	5%	4%
	Based on 1000 replicates of size n=20,000
P(y1(xe = ln(4)) = 1) − P(y1(xe = ln(2)) = 1)	0%	20%	28%	0%
P(y1(xe = ln(4)) = 1) − P(y1(xe = ln(2)) = 1)	0%	67%	29%	0%
P(y1(xe = ln(4)) = 1) − P(y1(xe = ln(2)) = 1)	0%	367%	37%	0%
P(y1(xe = ln(4)) = 1) − P(y1(xe = ln(2)) = 1)	0%	191%	8%	0%

The value in a particular cell of the table is the average percentage absolute bias, over the 1000 simulated samples, for a particular (estimator-q, average effect-t, sample size-j) combination, and is measured as

$$\left(\sum _{\text{m}=1}^{1000}\frac{1}{1000}\frac{\text{abs}(\text{AE}{(\text{t})}_{\text{qrm}}-\text{AE}(\text{t}))}{\text{abs}(\text{AE}(\text{t}))}\right)\times 100\%$$

where AE(t) denotes the true value of the t^th effect, AE(t)_qrm is its estimated value obtained by applying the q^th method to m^th sample of the r^th sample size, and

$$\begin{array}{l}\text{q}=\text{true MLE},\text{ na}\ddot{\text{i}}\text{ve},\text{ 2SPS},\text{ 2SRI}\\ \text{t}={\text{P(y}}_{{\text{1(x}}_{\text{e}}=\text{ln(4))}}=\text{1)}-{\text{P(y}}_{{\text{1(x}}_{\text{e}}=\text{ln(2))}}=\text{1)},\\ {\text{P(y}}_{{\text{2(x}}_{\text{e}}=\text{ln(4))}}=\text{1)}-{\text{P(y}}_{{\text{2(x}}_{\text{e}}=\text{ln(2))}}=\text{1)},\\ {\text{P(y}}_{{\text{3(x}}_{\text{e}}=\text{ln(4))}}=\text{1)}-{\text{P(y}}_{{\text{3(x}}_{\text{e}}=\text{ln(2))}}=\text{1)},\\ {\text{P(y}}_{{\text{4(x}}_{\text{e}}=\text{ln(4))}}=\text{1)}-{\text{P(y}}_{{\text{4(x}}_{\text{e}}=\text{ln(2))}}=\text{1)}\\ \text{r}=10000,20000.\end{array}$$