I can calculate the motion of heavenly bodies but not the madness of people. -Isaac Newton

## Estimating Time Series Models for Count Outcomes with SAS

In SAS, there is no out-of-box procedure to estimate time series models for count outcomes, which is similar to the one shown here (https://statcompute.wordpress.com/2015/03/31/modeling-count-time-series-with-tscount-package). However, as long as we understand the likelihood function of Poisson distribution, it is straightforward to estimate a time series model with PROC MODEL in the ETS module.

Below is a demonstration of how to estimate a Poisson time series model with the identity link function. As shown, the parameter estimates with related inferences are extremely close to the ones estimated with tscount() in R.

```data polio;
idx + 1;
input y @@;
datalines;
0  1  0  0  1  3  9  2  3  5  3  5  2  2  0  1  0  1  3  3  2  1  1  5  0
3  1  0  1  4  0  0  1  6 14  1  1  0  0  1  1  1  1  0  1  0  1  0  1  0
1  0  1  0  1  0  1  0  0  2  0  1  0  1  0  0  1  2  0  0  1  2  0  3  1
1  0  2  0  4  0  2  1  1  1  1  0  1  1  0  2  1  3  1  2  4  0  0  0  1
0  1  0  2  2  4  2  3  3  0  0  2  7  8  2  4  1  1  2  4  0  1  1  1  3
0  0  0  0  1  0  1  1  0  0  0  0  0  1  2  0  2  0  0  0  1  0  1  0  1
0  2  0  0  1  2  0  1  0  0  0  1  2  1  0  1  3  6
;
run;

proc model data = polio;
parms b0 = 0.5 b1 = 0.1 b2 = 0.1;
yhat = b0 + b1 * zlag1(y) + b2 * zlag1(yhat);
y = yhat;
lk = exp(-yhat) * (yhat ** y) / fact(y);
ll = -log(lk);
errormodel y ~ general(ll);
fit y / fiml converge = 1e-8;
run;

/* OUTPUT:
Nonlinear Liklhood Summary of Residual Errors

Equation       Model   Error        SSE        MSE   R-Square      R-Sq
y                  3     165      532.6     3.2277     0.0901    0.0791

Nonlinear Liklhood Parameter Estimates

Approx                  Approx
Parameter       Estimate     Std Err    t Value     Pr > |t|
b0              0.606313      0.1680       3.61       0.0004
b1              0.349495      0.0690       5.06       <.0001
b2              0.206877      0.1397       1.48       0.1405

Number of Observations       Statistics for System

Used               168    Log Likelihood    -278.6615
Missing              0
*/
```