  We present a broad class of semi-parametric models for time series of random
sums of positive variables. Our methodology allows the number of terms inside
the sum to be time-varying and is therefore well suited to many examples
encountered in the natural sciences. We study the stability properties of the
models and provide a valid statistical inference procedure to estimate the
model parameters. It is shown that the proposed quasi-maximum likelihood
estimator is consistent and asymptotically normally distributed. This work is
complemented by simulation results and applied to annual growth rate time
series of white spruce (Picea glauca) trees from a few dozen sites in Quebec
spanning 41 years, including one major spruce budworm (Choristoneura
fumiferana) outbreak from around 1968 to 1991. We found significant growth
reductions due to budworm-induced by defoliation up to two years in the past.
Our results also revealed positive effects of maximum temperature,
precipitation and the climate moisture index in the summer, as well as negative
effects of the climate moisture index in the spring and the maximum temperature
in the previous summer. However, considering the interaction of climate and
defoliation on growth did not improve the model's performance on this dataset.
This study represent a major advances and our result represent an useful tool
in the understanding of the combined effects of climate and insect defoliation
on tree growth in the face of climate change, where the frequency and the
severity of outbreaks, as well as an increase of temperature is expected.
