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simqi

Format:

     simqi [, pv genpv(newvar)
              ev genev(newvar)
              pr prval(value1 value2...) genpr(newvar1 newvar2...)
              fd(existing option) changex(var1 val1 val2 [& var2 val1 val2] )
               msims(#) tfunc(function) level(#) listx ]

Description:

After simulating parameters from the last estimation (see Section 8.1) and setting values for the explanatory variables (see Section 8.2), use simqi to simulate various quantities of interest, including predicted values, expected values, and first differences.

Predicted values contain two forms of uncertainty: ``fundamental'' uncertainty arising from sheer randomness in the world, and ``estimation'' uncertainty caused by not having an infinite number of observations. More technically, predicted values are random draws of the dependent variable from the stochastic component of the statistical model, given a random draw from the posterior distribution of the unknown parameters.

If there were no estimation uncertainty, the expected value would be a single number representing the mean of the distribution of predicted values. But estimates are never certain, so the the expected value must be a distribution rather than a point. To obtain this distribution, we average-away the fundamental variability, leaving only estimation uncertainty. For this reason, expected values have a smaller variance than predicted values, even though the point estimate should be roughly the same in both cases. simqi calculates two kinds of expected values: the expected value of $Y$, and the probability that $Y$ takes on a particular value. For models in which these two quantities are equal, simqi avoids redundancy by reporting only the probabilities.

Note: simulated expected values are equivalent to simulated probabilities for all the discrete choice models that simqi supports (logit, probit, ologit, oprobit, mlogit). In these models, the expected value of $Y$ is a vector, with each element indicating the probability that $Y=j$. Consider an ordered probit with outcomes 1, 2, 3. The expected value is [Pr($Y=1$), Pr($Y=2$), Pr($Y=3$)], the mean of a multinomial distribution that generates the dependent variable.

A first difference is the difference between two expected values. To simulate first differences use the fd ``wrapper'', which is described below.

simqi can generate predicted values, expected values and first differences for all the models that it supports. By default, however, it will only report the quantities of interest that appear in the table below. To view other quantities of interest or save the simulated quantities as new variables that can be analyzed and graphed, use one of simqi's options.

Statistical	Quantities displayed
Model	by default
regress	E(Y)
logit	Pr(Y=1)
probit	Pr(Y=1)
ologit	Pr(Y=j) for all j
oprobit	Pr(Y=j) for all j
mlogit	Pr(Y=j) for all j
poisson	E(Y)
nbreg	E(Y)
sureg	E($Y_{j}$) for all equations j
weibull	E(Y)

Options:

pv

displays a summary of the predicted values that simqi generated via simulation.

genpv(newvar)

saves the predicted values as a new variable in the current dataset. Each ``observation'' of newvar represents one simulated predicted value.

pr

displays a summary of the probabilities that simqi generated via simulation.

prval(value1 value2 ...)

instructs simqi to evaluate the probability that the dependent variable takes-on each of the listed values

genpr(newvar1 newvar2 ...)

saves the simulated probabilities as new variables in the current dataset. Each new ``observation'' represents one simulated probability. If both the prval() option and the genpr() option are used, simqi will save Pr(Y=value1) in $newvar1$, Pr(Y=value2) in $newvar2$, etc. If the prval() option is not specified, genpr() will save the probabilities in the order that they appear on the screen.

ev

displays a summary of expected values that simqi generated via simulation. This option is not available for discrete choice models, where it is redundant with pr.

genev(newvar)

saves the expected values in a new variable called newvar. Each observation of newvar represents one simulated expected value. This option is not available for discrete choice models, where it is redundant with genpr().

fd(existing option)

is a wrapper that makes it easy to simulate first differences. Simply wrap the fd() wrapper around an existing option and specify the changex() option.

changex(var1 val1 val2)

specifies how the explanatory variables (the $x$'s) should change when evaluating a first difference. changex uses the same basic syntax as setx, except that each explanatory variable has two values: a starting value and an ending value. For instance, fd(ev) changex(x1 .2 .8) instructs simqi to simulate a change in the expected value of $Y$ caused by increasing $x1$ from its starting value, 0.2, to its ending value, to 0.8.

level(#)

specifies the confidence level, in percent, for confidence intervals. The default is level(95) or the value set by set level. For more information on the set level command, see the on-line help for level.

msims(#)

sets the number of simulations to be used when calculating expected values. The number must be a positive integer. By default, the value of msims is set at 1000. simqi disregards the msims option whenever the expected value is parametrically defined.

listx

instructs simqi to list the x-values that were used to produce the quantities of interest. These values were set using the setx command.

tfunc(function)

allows the user to specify a transformation function for transforming the dependent variable. This option is only available for regress and sureg. The currently supported functions are

Function	Transformation (for all variables j)
squared	$y_{j} \longrightarrow y_{j} * y_{j}$
sqrt	$y_{j} \longrightarrow \sqrt(y_{j})$
exp	$y_{j} \longrightarrow e^{y_{j}}$
ln	$y_{j} \longrightarrow \ln(y_{j})$
logiti	$y_{j} \longrightarrow e^{y_{j}}/(1+\sum_j e^{y_{j}})$

Basic Examples:

To display the default quantities of interest for the last estimated model, type:

     . simqi

For a summary of the simulated expected values, type:

     . simqi, ev

For a summary of the simulated probabilities, Pr(Y=j), for all j categories of the dependent variable, type:

     . simqi, pr

To display only a summary of Pr(Y=1), the probability that the dependent variable takes on a value of 1, type:

     . simqi, prval(1)

To generate first differences, use the fd() wrapper and the changex() option. For instance, the following command will simulate the change in the expected value of Y caused by increasing $x4$ from 3 to 7, while holding other explanatory variables at their means

     . setx mean
     . simqi, fd(ev) changex(x4 3 7)

To simulate the change in the simulated probabilities, Pr(Y=j), for all j categories of the dependent variable, given an increase in $x4$ from its minimum to its mean, type:

     . setx mean
     . simqi, fd(pr) changex(x4 min mean)

If you are only interested in the change in Pr(Y=1) caused by raising x4 from its 20th to its 80th percentile when other variables are held at their mean, type:

     . setx mean
     . simqi, fd(prval(1)) changex(x4 p20 p80)

More Intricate Examples:

To display not only the simulated expected values but also the x-values used to produce them, we would type:

     . simqi, ev listx

simqi displays 95% confidence intervals by default, but we could modify the previous example to give a 90% confidence interval for the expected value:

     . simqi, ev listx level(90)

To save the simulated expected values in a new variable called $predval$, type:

     . simqi, genev(predval)

To simulate Pr(Y=0), Pr(Y=3), and Pr(Y=4), and then save the simulated probabilities as variables called $simpr0$, $simpr3$ and $simpr4$, type:

     . simqi, prval(0 3 4) genpr(simpr0 simpr3 simpr4)

The changex option can be arbitrarily complicated. Suppose that we want to simulate the change in Pr(Y=1) caused by simultaneously increasing $x1$ from .2 to .8 and $x2$ from ln(7) to ln(10). The following lines will produce the quantities we seek:

     . setx mean
     . simqi, fd(prval(1)) changex(x1 .2 .8 x2 ln(7) ln(10))

We could augment the previous example by requesting a second first difference, caused by increasing $x3$ from its median to its 90th percentile. Simply separate the two changex requests with an ampersand.

     . setx mean
     . simqi, fd(prval(1)) changex(x1 .2 .8 x2 ln(7) ln(10) & x3 median p90)

Likewise, the fd() option can be as intricate as we would like. For instance, suppose that we have run a poisson regression. We want to see what happens to Pr(Y=2), Pr(Y=3), and the expected count when we increase $x1$ from its minimum to its maximum. To obtain our quantities of interest, we would type:

     . setx mean
     . simqi, fd(prval(2 3)) fd(ev) changex(x1 min max)

simqi allows us to save any simulated variable for subsequent analysis. To find the mean, standard deviation, and a confidence interval around any quantity of interest that has been saved in memory, use the sumqi command. To graph the simulations, use graph or kdensity.

The tfunc() option reverses common transformations that users have applied to the dependent variable. Suppose that you have taken the log of the dependent variable before running estsimp regress. The command simqi would provide quantities of interest on the logged scale. If you wanted to reverse the transformation, thereby recovering the original scale, you could type

. simqi, tfunc(exp)