The HPQLIM Procedure

Output to SAS Data Set

Subsections:

XBeta, Predicted, and Residual

Xbeta is the structural part on the right-hand side of the model. The predicted value is the predicted dependent variable value. For censored variables, if the predicted value is outside the boundaries, it is reported as the closest boundary. The residual is defined only for continuous variables and is defined as

\[ \text {Residual} = \text {Observed} - \text {Predicted} \]

Error Standard Deviation

The error standard deviation is $\sigma _ i$ in the model. It varies only when the HETERO statement is used.

Marginal Effects

A marginal effect is defined as a contribution of one control variable to the response variable. For a binary choice model with two response categories, $\mu _{0}=-\infty $ and $\mu _{1}=0$, $\mu _{2}=\infty $. For an ordinal response model with M response categories ($\mu _{0},\cdots ,\mu _{M}$), define

\[ R_{i,j} = \mu _{j} - \mb{x}_{i}’\bbeta \]

The probability that the unobserved dependent variable is contained in the jth category can be written as

\[ P[\mu _{j-1}< y_{i}^{*} \leq \mu _{j}] = F(R_{i,j}) - F(R_{i,j-1}) \]

The marginal effect of changes in the regressors on the probability of $y_{i}=j$ is then

\[ \frac{\partial \text {Prob}[y_{i}=j]}{\partial \mb{x}} = [f(\mu _{j-1} - \mb{x}_{i}’\bbeta ) - f(\mu _{j} - \mb{x}_{i}’\bbeta )] \bbeta \]

where $f(x) = \frac{d F(x)}{dx}$. In particular,

\[ f(x) = \frac{d F(x)}{dx} = \left\{ \begin{array}{ll} \frac{1}{\sqrt {2\pi }}e^{-x^2/2} & \mr{(probit)} \\ \frac{e^{-x}}{[1+e^{(-x)}]^2} & \mr{(logit)} \end{array} \right. \]

The marginal effects in the truncated regression model are

\[ \frac{\partial {E}[y_{i}|L_{i} < y_{i}^{*} < R_{i}]}{\partial \mb{x}} = \bbeta \left[ 1 - \frac{(\phi (a_{i})-\phi (b_{i}))^2}{(\Phi (b_{i})-\Phi (a_{i}))^2} + \frac{a_{i}\phi (a_{i})-b_{i}\phi (b_{i})}{\Phi (b_{i})-\Phi (a_{i})} \right] \]

where $a_{i}=\frac{L_{i}-\mb{x}_{i}'\bbeta }{\sigma _ i}$ and $b_{i}=\frac{R_{i}-\mb{x}_{i}'\bbeta }{\sigma _ i}$.

The marginal effects in the censored regression model are

\[ \frac{\partial {E}[y|\mb{x}_{i}]}{\partial \mb{x}} = \bbeta \times \text {Prob}[L_{i}<y_{i}^{*}<R_{i}] \]

Expected and Conditionally Expected Values

The expected value is the unconditional expectation of the dependent variable. For a censored variable, it is

\[ E[y_ i] = \Phi (a_{i}) L_{i} + (\mb{x}_{i}’\bbeta + \lambda \sigma _ i) (\Phi (b_{i})-\Phi (a_{i})) + (1-\Phi (b_{i})) R_{i} \]

For a left-censored variable ($R_ i=\infty $), this formula is

\[ E[y_ i] = \Phi (a_{i}) L_{i} + (\mb{x}_{i}’\bbeta + \lambda \sigma _ i) (1-\Phi (a_{i})) \]

where $\lambda = \frac{\phi (a_{i})}{1-\Phi (a_{i})}$.

For a right-censored variable ($L_ i=-\infty $), this formula is

\[ E[y_ i] = (\mb{x}_{i}’\bbeta + \lambda \sigma _ i) \Phi (b_{i}) + (1-\Phi (b_{i})) R_{i} \]

where $\lambda = -\frac{\phi (b_{i})}{\Phi (b_{i})}$.

For a noncensored variable, this formula is

\[ E[y_ i] = \mb{x}_{i}’\bbeta \]

The conditional expected value is the expectation when the variable is inside the boundaries:

\[ E[y_ i| L_ i< y_ i < R_ i] = \mb{x}_{i}’\bbeta + \lambda \sigma _ i \]

Technical Efficiency

Technical efficiency for each producer is computed only for stochastic frontier models.

In general, the stochastic production frontier can be written as

\[ y_ i=f(x_ i;\beta )\exp \{ v_ i\} TE_ i \]

where $y_ i$ denotes producer i’s actual output, $f(\cdot )$ is the deterministic part of the production frontier, $\exp \{ v_ i\} $ is a producer-specific error term, and $TE_ i$ is the technical efficiency coefficient, which can be written as

\[ TE_ i=\frac{y_ i}{f(x_ i;\beta )\exp \{ v_ i\} } \]

For a Cobb-Douglas production function, $TE_ i=\exp \{ -u_ i\} $. For more information, see the section Stochastic Frontier Production and Cost Models.

The cost frontier can be written in general as

\[ E_ i=c(y_ i,w_ i;\beta )\exp \{ v_ i\} /CE_ i \]

where $w_ i$ denotes producer i’s input prices, $c(\cdot )$ is the deterministic part of the cost frontier, $\exp \{ v_ i\} $ is a producer-specific error term, and $CE_ i$ is the cost efficiency coefficient, which can be written as

\[ CE_ i=\frac{c(x_ i,w_ i;\beta )\exp \{ v_ i\} }{E_ i} \]

For a Cobb-Douglas cost function, $CE_ i=\exp \{ -u_ i\} $. For more information, see the section Stochastic Frontier Production and Cost Models. Hence, both technical and cost efficiency coefficients are the same. The estimates of technical efficiency are provided in the following subsections.

Normal-Half-Normal Model

Define $\mu _*=-\epsilon \sigma _ u^2/\sigma ^2$ and $\sigma _*^2=\sigma _ u^2\sigma _ v^2/\sigma ^2$. Then, as shown by Jondrow et al. (1982), conditional density is as follows:

\[ f(u|\epsilon )=\frac{f(u,\epsilon )}{f(\epsilon )} =\frac{1}{\sqrt {2\pi }\sigma _*}\exp \left\{ -\frac{(u-\mu _*)^2}{2\sigma _*^2}\right\} \bigg/\left[1-\Phi \left(-\frac{\mu _*}{\sigma _*}\right)\right] \]

Hence, $f(u|\epsilon )$ is the density for $N^+(\mu _*,\sigma _*^2)$.

From this result, it follows that the estimate of technical efficiency (Battese and Coelli 1988) is

\[ TE1_ i=E(\exp \{ -u_ i\} |\epsilon _ i)=\left[\frac{1-\Phi (\sigma _*-\mu _{*i}/\sigma _*)}{1-\Phi (-\mu _{*i}/\sigma _*)}\right]\exp \left\{ -\mu _{*i}+\frac{1}{2}\sigma _*^2\right\} \]

The second version of the estimate (Jondrow et al. 1982) is

\[ TE2_ i=\exp \{ -E(u_ i|\epsilon _ i)\} \]

where

\[ E(u_ i|\epsilon _ i)=\mu _{*i}+\sigma _*\left[\frac{\phi (-\mu _{*i}/\sigma _*)}{1-\Phi (-\mu _{*i}/\sigma _*)}\right]=\sigma _*\left[\frac{\phi (\epsilon _ i\lambda /\sigma )}{1-\Phi (\epsilon _ i\lambda /\sigma )}-\left(\frac{\epsilon _ i\lambda }{\sigma }\right)\right] \]

Normal-Exponential Model

Define $ A=-\tilde{\mu }/\sigma _ v$ and $\tilde{\mu }=-\epsilon -\sigma _ v^2/\sigma _ u$. Then, as shown by Kumbhakar and Lovell (2000), conditional density is as follows:

\[ f(u|\epsilon ) =\frac{1}{\sqrt {2\pi }\sigma _ v\Phi (-\tilde{\mu }/\sigma _ v)}\exp \left\{ -\frac{(u-\tilde{\mu })^2}{2\sigma ^2}\right\} \]

Hence, $f(u|\epsilon )$ is the density for $N^+(\tilde{\mu },\sigma _ v^2)$.

From this result, it follows that the estimate of technical efficiency is

\[ TE1_ i=E(\exp \{ -u_ i\} |\epsilon _ i)=\left[\frac{1-\Phi (\sigma _ v-\tilde{\mu }_ i/\sigma _ v)}{1-\Phi (-\tilde{\mu }_ i/\sigma _ v)}\right]\exp \left\{ -\tilde{\mu }_ i+\frac{1}{2}\sigma _ v^2\right\} \]

The second version of the estimate is

\[ TE2_ i=\exp \{ -E(u_ i|\epsilon _ i)\} \]

where

\[ E(u_ i|\epsilon _ i)=\tilde{\mu }_ i+\sigma _ v\left[\frac{\phi (-\tilde{\mu }_ i/\sigma _ v)}{1-\Phi (-\tilde{\mu }_ i/\sigma _ v)}\right]=\sigma _ v\left[\frac{\phi (A)}{\Phi (-A)}-A\right] \]

Normal–Truncated Normal Model

Define $\tilde{\mu }=(-\sigma _ u^2\epsilon _ i+\mu \sigma _ v^2)/\sigma ^2$ and $\sigma _*^2=\sigma _ u^2\sigma _ v^2/\sigma ^2$. Then, as shown by Kumbhakar and Lovell (2000), conditional density is as follows:

\[ f(u|\epsilon )= \frac{1}{\sqrt {2\pi }\sigma _*[1-\Phi (-\tilde{\mu }/\sigma _*)]}\exp \left\{ -\frac{(u-\tilde{\mu })^2}{2\sigma _*^2}\right\} \]

Hence, $f(u|\epsilon )$ is the density for $N^+(\tilde{\mu },\sigma _*^2)$.

From this result, it follows that the estimate of technical efficiency is

\[ TE1_ i=E(\exp \{ -u_ i\} |\epsilon _ i)=\frac{1-\Phi (\sigma _*-\tilde{\mu }_ i/\sigma _*)}{1-\Phi (-\tilde{\mu }_ i/\sigma _*)}\exp \left\{ -\tilde{\mu }_ i+\frac{1}{2}\sigma _*^2\right\} \]

The second version of the estimate is

\[ TE2_ i=\exp \{ -E(u_ i|\epsilon _ i)\} \]

where

\[ E(u_ i|\epsilon _ i)=\tilde{\mu }_ i+\sigma _*\left[\frac{\phi (\tilde{\mu }_ i/\sigma _*)}{1-\Phi (-\tilde{\mu }_ i/\sigma _*)}\right] \]