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Wechsung, Maximilian (D-CHAR-BCE) ; Neumann, Michael H. (D-FSU-MI)
Consistency of a nonparametric least squares estimator in integer-valued
GARCH models. (English summary)J. Nonparametr. Stat. 34 (2022), no. 2, 491–519.

In this article, the convergence rate of a least squares estimator of a nonparametricinteger-valued GARCH (INGARCH) model is explored. An INGARCH process can berepresented as a bivariate one {(λt , Yt )}, where a time series of count variables {Yt }obeys a conditional Poisson distribution and the intensity {λt } is modeled as a recursiverelation.
A nonparametric version of INGARCH processes is studied in this article. Let Ft :=σ{λs , Ys : s ≤ t} be the σ-field generated by the process up to time t. Also, let m be an
unknown, nonnegative link function. Then, the INGARCH process concerned is modeledas
Yt | Ft−1 ∼ P oiss (λt ) and λt = m (Yt−1 , . . . , Yt−p , λt−1 , . . . , λt−q ) .
In this model, the process {Yt } obeys a nonparametric INGARCH(p, q) count processwith accompanying intensity process {λt }.
A particular focus is on an INGARCH(1, 1) process with the intensity λt =m(λt−1 , Yt−1 ). A least squares estimation of the unknown function m is investigated.
Existence of a stationarity distribution π of the bivariate process {(λt , Yt )} is ensured if
m is bounded in λ and smooth (or contractive, to be more precise) in both arguments.
To approximate m, consider a class of function g [i] that satisfiesnog [0] (λ, Yt ) = g (λ, Yt ) and g [k] (λ, Yt , . . . , Yt+k ) = g g [k−1] (λ, Yt , . . . , Yt+k−1 ) , Yt+k .
Then, for given (n + 1) consecutive observations {Yt }nt=0 , the least squares estimator ofm, denoted bym
b n , can be obtained as g [i] that minimizes the empirical contrast funcPn−1tional (1/n) i=0 {Yi+1 − g [i] (0, Y0 , . . . , Yi )}2 . Consistency of mb n for m is establishedby finding the convergence rate of its L2 (π) lossZ2L (mb n , m) := {mb n (λ, y) − m (λ, y)} π (dλ, dy) = Op n−2/3 log2 n .
It is conjectured that the optimal rate of the L2 (π) loss would be of order n−2/3 . The
above convergence rate appears to be nearly optimal in light of this conjecture.
The convergence result documents a theoretical property of mb n and does not provide
its specific form. Sieve methods are recommended for the practical implementation ofmb n . The article also refers to the possibility of estimating the link function m for general
Masayuki Hirukawa
INGARCH(p, q) processes.
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