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The example below shows how to estimate a simple univariate Poisson time series model with the tscount package. While the model estimation is straightforward and yeilds very similar parameter estimates to the ones generated with the acp package (https://statcompute.wordpress.com/2015/03/29/autoregressive-conditional-poisson-model-i), the prediction mechanism is a bit tricky.

1) For the in-sample and the 1-step-ahead predictions:

yhat_[t] = beta0 + beta1 * y_[t – 1] + beta2 * yhat_[t – 1]

2) For the out-of-sample predictions with the observed Y unavailable:

yhat_[t] = beta0 + beta1 * yhat_[t – 1] + beta2 * yhat_[t – 1]

```library(tscount)

mdl <- tsglm(cnt\$y, model = list(past_obs = 1, past_mean = 1), distr = "poisson")
summary(mdl)
# tsglm(ts = cnt\$y, model = list(past_obs = 1, past_mean = 1),
#     distr = "poisson")
#
# Coefficients:
#              Estimate  Std. Error
# (Intercept)     0.632      0.1774
# beta_1          0.350      0.0687
# alpha_1         0.184      0.1455
# Standard errors obtained by normal approximation.
#
# Distribution family: poisson
# Number of coefficients: 3
# Log-likelihood: -279.2738
# AIC: 564.5476
# BIC: 573.9195

### in-sample prediction ###
cnt\$yhat <- mdl\$fitted.values
tail(cnt, 3)
#     y      yhat
# 166 1 0.8637023
# 167 3 1.1404714
# 168 6 1.8918651

### manually check ###
beta <- mdl\$coefficients
pv167 <- beta[1] + beta[2] * cnt\$y[166] + beta[3] * cnt\$yhat[166]
#  1.140471
pv168 <- beta[1] + beta[2] * cnt\$y[167] + beta[3] * cnt\$yhat[167]
#  1.891865

### out-of-sample prediction ###
oot <- predict(mdl, n.ahead = 3)
# [1] 3.080667 2.276211 1.846767

### manually check ###
ov2 <- beta[1] + beta[2] * oot[[1]][1] + beta[3] * oot[[1]][1]
#  2.276211
ov3 <- beta[1] + beta[2] * oot[[1]][2] + beta[3] * oot[[1]][2]
#  1.846767
```