2 FRAMEWORK

We are given n observations, $Z_{n,t}\in \left ( \mathcal {Z},\left \Vert \cdot \right \Vert \right ) $ , $t=1,\ldots ,n$ , where $\left ( \mathcal {Z} ,\left \Vert \cdot \right \Vert \right ) $ is a Banach space, from a time-series model characterized by a finite-dimensional vector of unknown parameters $\theta \in \Theta \subset \mathbb {R}^{d_{\theta }}$ to be estimated. In most applications, $Z_{n,t}$ will also be finite-dimensional but our theory allows for, for example, functional data as well. We take as given an objective function $\ell _{n,t}\left ( \theta \right ) =\ell \left ( \mathcal {Z}_{n,t};\theta \right ) \in \mathbb {R}$ , where $\mathcal {Z} _{n,t}=\left ( Z_{n,t},\dots Z_{n,0},Z_{n,-1},\dots \right ) $ . Since we do not observe the process before $t=1$ , we here initialize the process at deterministic values chosen by us, $Z_{n,-t}=z_{-t}$ , $t\geq 1$ . Under regularity conditions stated below, the effect of the initial values will vanish asymptotically.

The objective function is assumed to identify the data-generating parameter as its maximizer, $\theta =\arg \max _{\theta ^{\prime }\in \Theta }\mathbb {E} \left [ \ell _{n,t}\left ( \theta ^{\prime }\right ) \right ] $ , if $\theta $ indeed was time-invariant, and $\ell _{n,t}\left ( \theta ^{\prime }\right ) $ was stationary and ergodic. In this case, the natural estimator is to replace population expectations by sample ones and estimate $\theta $ by $ \hat {\theta }=\arg \max _{\theta ^{\prime }\in \Theta }\frac {1}{n} \sum _{t=1}^{n}\ell _{n,t}\left ( \theta ^{\prime }\right ) $ .

Suppose now that in fact $\theta $ is varying over time so that $\mathcal {Z} _{n,t}$ is generated by $\theta _{n,t}=\theta (t/n)$ , $t=1,\ldots ,n$ , where $ \theta :\left [ 0,1\right ] \mapsto \Theta $ is an unknown function that characterizes the time variation in the parameters.Footnote ¹ At the same time, the objective function is still assumed to identify the parameter in the sense that $\theta \left ( t/n\right ) =\arg \max _{\theta ^{\prime }\in \Theta }\mathbb {E}\left [ \ell _{n,t}\left ( \theta ^{\prime }\right ) \right ] $ . We then propose to estimate $\theta \left ( u\right ) $ at any given value $u\in \left [ 0,1\right ] $ using local polynomial estimators: First, for $t/n$ in a neighborhood of u, we approximate $\theta \left ( t/n\right ) $ by the following polynomial of order $m\geq 0$ :

(1) $$ \begin{align} \theta _{u,\beta }^{\ast }\left( t/n\right) :=\beta _{1}+\beta _{2}\left( t/n-u\right) +\cdots +\beta _{m+1}\left( t/n-u\right) ^{m}/m!=D_{m}\left( t/n-u\right) \beta, \end{align} $$

where $\beta =\left ( \beta _{1}^{\prime },\ldots ,\beta _{m+1}^{\prime }\right ) ^{\prime }\in \mathbb {R}^{\left ( m+1\right ) d_{\theta }}$ with $\beta _{i+1}=\theta ^{\left ( i\right ) }\left ( u\right ) =\partial ^{i}\theta \left ( u\right ) /\partial u^{i}\in \mathbb {R}^{d_{\theta }}$ and

$$ \begin{align*} D_{m}\left( u\right) =\left( 1,u,u^{2}/2\ldots,u^{m}/m!\right) \otimes I_{d_{\theta }}\in \mathbb{R}^{d_{\theta }\times \left( m+1\right) d_{\theta }}. \end{align*} $$

Next, to control the approximation error, $\theta \left ( t/n\right ) -\theta _{u,\beta }^{\ast }\left ( t/n\right ) $ , we introduce a kernel weighted version of the “global” objective function evaluated at the polynomial approximation:

$$ \begin{align*} L_{n}\left( \beta |u\right) =\frac{1}{n}\sum_{t=1}^{n}K_{b}\left( t/n-u\right) \ell _{n,t}\left( \theta _{u,\beta }^{\ast }\left( t/n\right) \right), \end{align*} $$

where $K_{b}\left ( \cdot \right ) =K\left ( \cdot /b\right ) /b$ , $K:\mathbb {R} \mapsto \mathbb {R}$ is a kernel function, and $b=b_{n}>0$ is a bandwidth. The kernel weights ensure that when $t/n-u$ is “large,” the corresponding observations are down weighted in the estimation, thereby controlling for the aforementioned approximation error. We then estimate the polynomial coefficients by

(2) $$ \begin{align} \hat{\beta}\left( u\right) =\arg \max_{\beta \in \mathcal{B}}L_{n}\left( \beta |u\right), \end{align} $$

where

$$ \begin{align*} \mathcal{B}=\left\{ \beta \in \mathbb{R}^{\left( m+1\right) d_{\theta }}:\theta _{u,\beta }^{\ast }\left( v\right) \in \Theta \text{ for }v\in \left[ 0,1\right] \right\}. \end{align*} $$

The estimated $\beta $ coefficients are used as estimates of $\theta \left ( u\right ) $ and its first m derivatives, $\hat {\theta }^{\left ( i\right ) }\left ( u\right ) =\hat {\beta }_{i+1}\left ( u\right ) $ , $i=0,\ldots ,m$ . When $m=0$ , we recover the standard local-constant estimator. The above class of estimators is similar to the ones considered in Fan et al. (Reference Fan, Heckman and Wand1995) for so-called varying-coefficient models, except that we consider time-series models with the “regressor” that we smooth over being normalized time, $t/n$ , and do not restrict $\theta \mapsto \ell _{n,t}\left ( \theta \right ) $ to be convex.

The choice of the order of the polynomial, m, should reflect the degree of smoothness that we are willing to assume $u\mapsto \theta \left ( u\right ) $ has. If $\theta \left ( u\right ) $ is m times differentiable, then we should use this m in the estimation for optimal control of the bias in the nonparametric estimation. On the other hand, increasing the order of the polynomial tends to increase the variability of the resulting estimator, since more local parameters are introduced in the estimation. For a further discussion of this issue, we refer the reader to Section 3.3 of Fan and Gijbels (Reference Fan and Gijbels2018).

Our framework includes Markov processes and stochastic processes with infinite memory (see, e.g., Doukhan and Wintenberger, Reference Doukhan and Wintenberger2008; Bardet et al., Reference Bardet, Doukhan and Wintenberger2022) as special cases. In Section 4, we apply our general theory to the following class of Markov models with exogenous covariates:

(3) $$ \begin{align} Y_{n,t}=G\left( Y_{n,t-1},\ldots,Y_{n,t-q},X_{n,t-1},\varepsilon _{t};\theta \left( t/n\right) \right), \end{align} $$

where $G:\mathcal {Y}^{q}\times \mathcal {X}\times \mathcal {E}\times \Theta $ is a known function, $X_{n,t-1}$ is a Markov process of exogenous covariates, and $ \varepsilon _{t}$ is a sequence of errors. For a given specification of G and the distribution of $\varepsilon _{t}$ , we can then derive the corresponding log-likelihood for the model with time-invariant parameters, $\ell _{n,t}\left ( \theta \right ) =\ell \left ( Z_{n,t},X_{n,t-1};\theta \right ) $ , where $Z_{n,t}=\left ( Y_{n,t},X_{n,t}\right ) $ . Below, we provide three examples of models that our theory applies to.

Example 1. Time-varying threshold autoregressive with exogenous covariates (tv-TAR-X) model with two regimes:

(4) $$ \begin{align} Y_{n,t}=\omega +\sum_{i=1}^{q}\alpha _{1,i}\left( t/n\right) Y_{n,t-i}^{+}+\sum_{i=1}^{q}\alpha _{2,i}\left( t/n\right) Y_{n,t-1}^{-}+\gamma \left( t/n\right) ^{\prime }X_{n,t-1}+\varepsilon _{t}, \end{align} $$

where $y^{+}:=\max \left \{ y,0\right \} $ and $y^{-}:=\min \left \{ y,0\right \} $ . Here, $X_{n,t-1}$ contains additional predictors, and $ \varepsilon _{t}$ is i.i.d. with $\mathbb {E}\left [ \varepsilon _{t}\right ] =0 $ and $\mathbb {E}\left [ \varepsilon _{t}^{2}\right ] <\infty $ . A natural estimator of $\theta =\left ( \omega ,\alpha _{1}^{\prime },\alpha _{2}^{\prime },\gamma ^{\prime }\right ) ^{\prime }$ is the least-squares one so that $\ell _{n,t}\left ( \theta \right ) =-\left ( Y_{n,t}-\omega -\sum _{i=1}^{q}\alpha _{1,i}Y_{n,t-1}^{+}-\sum _{i=1}^{q}\alpha _{2,i}Y_{n,t-1}^{-}-\gamma ^{\prime }X_{n,t-1}^{-}\right ) ^{2}$ .

Example 2. Time-varying ARCH model with covariates (tv-ARCH-X):

(5) $$ \begin{align} Y_{n,t}=\lambda _{n,t}\left( \theta \left( t/n\right) \right) \varepsilon _{t}^{2},\;\ \ \lambda _{n,t}\left( \theta \right) =\omega +\sum_{i=1}^{q}\alpha _{i}Y_{n,t-1}+\gamma ^{\prime }X_{n,t-1}, \end{align} $$

where $\theta =\left ( \omega ,\alpha ^{\prime },\gamma ^{\prime }\right ) ^{\prime }$ , $\varepsilon _{t}$ is i.i.d. with $\mathbb {E}\left [ \varepsilon _{t}^{2}\right ] =1,$ and $X_{n,t-1}$ contains additional predictors. The Gaussian log-likelihood function takes the form $\ell _{n,t}\left ( \theta \right ) =-Y_{n,t}/\lambda _{n,t}\left ( \theta \right ) -\log \left ( \lambda _{n,t}\left ( \theta \right ) \right ) $ , where $\theta =\left ( \omega ,\alpha ^{\prime },\gamma ^{\prime }\right ) ^{\prime }$ .

Example 3. Time-varying Poisson autoregression with exogenous covariates (tv-PARX):

(6) $$ \begin{align} Y_{n,t}|\mathcal{F}_{n,t-1}\sim \mathrm{Poisson}\left( \lambda _{n,t}\left( \theta \left( t/n\right) \right) \right) ,\;\ \ \lambda _{n,t}\left( \theta \right) =\omega +\!\sum_{i=1}^{q}\alpha _{i}Y_{n,t-1}+\gamma ^{\prime }X_{n,t-1}, \end{align} $$

where $\mathrm {Poisson}\left ( \lambda \right ) $ denotes the Poisson distribution with intensity parameter $\lambda $ , $\theta =\left ( \omega ,\alpha ^{\prime },\gamma ^{\prime }\right ) ^{\prime }$ , and $X_{n,t-1}$ contains additional predictors. The log-likelihood function takes the form $ \ell _{n,t}\left ( \theta \right ) =Y_{n,t}\log \left ( \lambda _{n,t}\left ( \theta \right ) \right ) -\lambda _{n,t}\left ( \theta \right ) $ .

3 ASYMPTOTIC THEORY

We here provide an asymptotic theory for $\hat {\beta }$ . One complication of this analysis is that the components of $\hat {\beta }$ converge with different rates. We follow the existing literature and handle this issue by introducing a re-scaled version of $\hat {\beta }$ (see, e.g., Han and Kristensen, Reference Han and Kristensen2014, for a similar approach): Define $\hat {\alpha }=U_{n}\hat {\beta }=(\hat {\theta } \left ( u\right ) ^{\prime },b\hat {\theta }^{\left ( 1\right ) }\left ( u\right ) ^{\prime },\ldots ,b^{m}\hat {\theta }^{\left ( m\right ) }\left ( u\right ) ^{\prime })^{\prime }$ , where

$$ \begin{align*} U_{n}=diag\left\{ 1,b,\ldots,b^{m}\right\} \otimes I_{d_{\theta }}\in \mathbb{R} ^{\left( m+1\right) d_{\theta }\times \left( m+1\right) d_{\theta }} \end{align*} $$

is a weighting matrix containing their relative convergence rates, Given that $U_{n}$ is non-singular, the estimation problem (2) is equivalent to solving

(7) $$ \begin{align} \hat{\alpha}=\arg \max_{\alpha \in \mathcal{A}}Q_{n}\left( \alpha |u\right) ,\hspace{6pt} Q_{n}\left( \alpha |u\right) =\frac{1}{n}\sum_{t=1}^{n}K_{b}\left( t/n-u\right) \ell _{n,t}\left( D_{m,b}\left( t/n-u\right) \alpha \right), \end{align} $$

where $D_{m,b}\left ( u\right ) =D_{m}\left ( u/b\right ) $ and

(8) $$ \begin{align} \mathcal{A}=\left\{ \alpha \in \mathbb{R}^{\left( m+1\right) d_{\theta }}:D_{m}\left( v\right) \alpha \in \Theta ,\ \forall v\in \mathcal{K}\right\} \end{align} $$

with $\mathcal {K}$ denoting the support of K. We will then analyze the properties of $\hat {\alpha }$ .

Due to the time-varying parameters, $Z_{n,t}$ will generally be non-stationary. To develop an asymptotic theory that allows for this feature, we will rely on the concept of local stationarity as introduced by Dahlhaus (Reference Dahlhaus1997); see also Dahlhaus and Subba Rao (Reference Dahlhaus and Subba Rao2006) and Dahlhaus et al. (Reference Dahlhaus, Richter and Wu2019). We first generalize this concept to sequences of random functions.

Definition 1. A triangular family of random sequences $W_{n,t}\left ( \theta \right ) $ , $\theta \in \Theta $ , $t=1,2,\ldots ,n$ and $n\geq 1$ , is uniformly locally stationary on $\Theta $ (ULS $\left (p,r,\Theta \right ) $ ) for some $ p,r>0$ if there exists a family of processes $W_{t}^{\ast }\left ( \theta |u\right ) $ , $u\in \left [ 0,1\right ] $ , such that: (i) the process $\left \{ W_{t}^{\ast }\left ( \theta |u\right ) \right \} $ is stationary and ergodic for all $\left ( \theta ,u\right ) \in \Theta \times \left [ 0,1\right ] $ and (ii) for some $C<\infty $ and $\rho <1$ ,

$$ \begin{align*} \mathbb{E}\left[ \sup_{\theta \in \Theta }\left\Vert W_{n,t}\left( \theta \right) -W_{t}^{\ast }\left( \theta |u\right) \right\Vert ^{p}\right] ^{1/p}\leq C\left( \left\vert \frac{t}{n}-u\right\vert ^{r}+\frac{1}{n^{r}} +\rho ^{t}\right). \end{align*} $$

If $W_{n,t}\left ( \theta \right ) =W_{n,t}$ does not depend on any parameters, we write LS $\left ( p,r\right ) $ . The above condition states that $W_{n,t}\left ( \theta \right ) $ may be non-stationary, but it is locally in time well-approximated by a stationary version $W_{t}^{\ast }\left ( \theta |u\right ) $ . Compared to existing definitions of local stationarity, we allow for an additional term $\rho ^{t}$ to appear in the approximation error. This is needed in order to allow for the initial value of $ W_{n,t}\left ( \theta \right ) $ to be chosen arbitrarily. In contrast, by not including $\rho ^{t}$ in their definitions, most of the existing literature implicitly assumes that $W_{n,t}\left ( \theta \right ) $ has been initialized at $W_{n,0}\left ( \theta \right ) =W_{0}^{\ast }\left ( \theta |u\right ) $ . When used in the analysis of local estimators, this latter definition implicitly requires that the DGP changes as the researcher varies u in the local log-likelihood which is a rather peculiar assumption. In contrast, the above definition allows for $W_{n,0}\left ( \theta \right ) $ to be initialized at a given fixed value—as long as the impact of this dies out with rate $\rho $ .

For an example of how the additional error term appears in autoregressive models, we refer the reader to the proof of Lemma 3 in Section 4 which allows for arbitrary initialization of the DGP. The additional error term due to different initializations is here assumed to decay geometrically and so our definition rules out long-memory-type processes. This is mostly for simplicity and we expect that most of our results can be generalized to allow for slower decay rates.

We will then require that $\ell _{n,t}\left ( \theta \right ) $ is ULS $\left ( p,r,\Theta \right ) $ with stationary approximation $\ell _{t}^{\ast }\left ( \theta |u\right ) =\ell \left ( \mathcal {Z}_{t}^{\ast }\left ( u\right ) ,\theta \right ) $ , where $\mathcal {Z}_{t}^{\ast }\left ( u\right ) =\left ( Z_{t}^{\ast }\left ( u\right ) ,\ Z_{t-1}^{\ast }\left ( u\right ),\dots \right ) $ is the stationary solution to the model being estimated when $\theta _{n,t}=\theta \left ( u\right ) $ is constant. To illustrate, consider again (3). Under regularity conditions on G and $\varepsilon _{t}$ (see Section 4 for details), the stationary solution will in this case take the form

$$ \begin{align*} Y_{t}^{\ast }\left( u\right) =G\left( Y_{t-1}^{\ast }\left( u\right) ,\ldots,Y_{t-q}^{\ast }\left( u\right) ,X_{t-1}^{\ast }\left( u\right) ,\, \varepsilon _{t};\ \theta \left( u\right) \right), \end{align*} $$

where we impose the high-level condition that the exogenous covariates are locally stationary. If the DGP is locally stationary, it follows under great generality that the likelihood and its derivatives are also locally stationary (cf. Section 4).

The next step in our proof is to establish a uniform law of large numbers (ULLN) for the stationary approximation of $Q_{n}\left ( \alpha |u\right ) $ , $ Q_{n}^{\ast }\left ( \alpha |u\right ) =\frac {1}{n}\sum _{t=1}^{n}K_{b}\left ( t/n-u\right ) \ell _{t}^{\ast }\left ( D_{b}\left ( t/n-u\right ) \alpha |u\right ) $ . A sufficient condition for a ULLN to hold is that $\theta \mapsto \ell _{n,t}^{\ast }\left ( \theta |u\right ) $ is $L_{p}$ -continuous.

Definition 2. A stationary process $W_{t}^{\ast }\left ( \theta |u\right ) $ is said to be $L_{p}$ -continuous w.r.t. $\theta $ if $ \mathbb {E}\left [ \left \Vert W_{t}^{\ast }\left ( \theta |u\right ) \right \Vert ^{p}\right ] <\infty $ for all $\theta \in \Theta $ and

$$ \begin{align*} \forall \epsilon>0\exists \delta >0:\mathbb{E}\left[ \sup_{\theta ^{\prime }:\left\Vert \theta -\theta ^{\prime }\right\Vert <\delta }\left\Vert W_{t}^{\ast }\left( \theta |u\right) -W_{t}^{\ast }\left( \theta ^{\prime }|u\right) \right\Vert ^{p}\right] ^{1/p}<\epsilon. \end{align*} $$

Imposing $L_{p}$ -continuity w.r.t. $\theta $ is weaker than almost surely continuity: If $\theta \mapsto W_{t}^{\ast }\left ( \theta |u\right ) $ is almost surely continuous with $\mathbb {E}\left [ \sup _{\theta \in \Theta }\left \Vert W_{t}^{\ast }\left ( \theta |u\right ) \right \Vert ^{p}\right ] <\infty , $ the process is also $L_{p}$ -continuous since $DW_{t}(\delta )=\sup _{\Vert \theta -\theta ^{\prime }\Vert \leq \delta }\left \Vert W_{t}^{\ast }\left ( \theta |u\right ) -W_{t}^{\ast }\left ( \theta ^{\prime }|u\right ) \right \Vert ^{p}$ , $\delta>0$ , will then satisfy $\lim _{\delta \rightarrow 0}DW_{t}(\delta )=0$ almost surely and so, by dominated convergence, $\lim _{\delta \rightarrow 0}\mathbb {E}[DW_{t}(\delta )]=0$ . It is easily verified that $L_{p}$ -continuity w.r.t. $\theta $ implies stochastic equicontinuity of $Q_{n}^{\ast }\left ( \alpha |u\right ) $ and so a ULLN holds (cf. Lemma A(i) in Appendix A.1).

We are now ready to state the regularity conditions used to show consistency.

Assumption 1(i) imposes stronger than usual assumptions on $ K$ and excludes, among others, the Gaussian kernel and higher-order kernels. It includes, on the other hand, the Epanechnikov and the triangular kernel. The restriction that $K\left ( \cdot \right ) \geq 0$ is used to ensure the identification of the parameters when $m>0$ ; without this, identification is not necessarily guaranteed; see below for further discussion. For the analysis of the local constant estimator ( $m=0$ ), all subsequent results will go through with K having full support and taking negative values.

The compact support assumption greatly simplifies our analysis of local polynomial estimation of non-concave models: In order to establish uniform convergence of the likelihood, we require $\Theta $ to be compact as is standard in the literature. But under this restriction, it is easily checked that $D_{m,b}\left ( v\right ) \alpha \notin \Theta $ as $b\rightarrow 0$ for any given $\alpha =\left ( \alpha _{1},\ldots ,\alpha _{m+1}\right ) $ with $ \alpha _{i}\neq 0$ for some $i\geq 1$ and any $v\neq 0$ . Thus, to allow for kernels with unbounded support, we would generally need the parameter space $ \mathcal {A}$ , as defined in (8), to collapse at $\left \{ \left ( \alpha _{1},0,\ldots ,0\right ) :\alpha _{1}\in \Theta \right \} $ as $ b\rightarrow 0$ . Such shrinking behavior in turn means that a formal Taylor expansion of $\ell _{n,t}\left ( D_{m,b}\left ( v\right ) \alpha \right ) $ w.r.t. $\alpha $ is difficult to obtain and so standard arguments to establish asymptotic normality of $\hat {\alpha }$ cannot be applied. On the other hand, by restricting the support $\mathcal {K}$ to be compact, $ K_{b}\left ( v\right ) \ell _{n,t}\left ( D_{m,b}\left ( v\right ) \alpha \right ) $ is well-defined for all $\alpha \in \mathcal {A}$ and $v\in \mathbb {R}$ (where we set $K_{b}\left ( v\right ) \ell _{n,t}\left ( D_{m,b}\left ( v\right ) \alpha \right ) =0$ for $v/b\notin \mathcal {K}$ ). Moreover, $\left ( \alpha _{1},0,\ldots ,0\right ) $ is an interior point of $\mathcal {A,}$ and so in our analysis of $\hat {\alpha ,}$ we can employ standard arguments involving a Taylor expansion of the score function around this point.

Assumption 2 is standard in the analysis of “global” extremum estimators of stationary models on the form $\tilde {\theta }\left ( u\right ) =\arg \max _{\theta \in \Theta }\sum _{t=1}^{n}\ell _{t}^{\ast }\left ( \theta |u\right ) /n$ . In particular, for a given time-series model, we can import existing results for verification of Assumption 2 (ii) and (iii); see Section 4 for more details. Assumption 2 in conjunction with $K\left ( \cdot \right ) \geq 0$ ensures that the local polynomial estimator identifies $\theta \left ( u\right ) $ . If we allow for kernels that take negative values, we have to replace Assumption 2(iii) with the following more abstract identification condition: The function $Q^{\ast }\left ( \alpha |u\right ) =\int K\left ( v\right ) \mathbb {E}\left [ \ell _{t}^{\ast }\left ( D_{m}\left ( v\right ) \alpha |u\right ) \right ] dv$ satisfies $Q^{\ast }\left ( \alpha |u\right ) <Q^{\ast }\left ( \left ( \theta \left ( u\right ) ,0,\ldots ,0\right ) |u\right ) $ for any $\alpha \neq \left ( \theta \left ( u\right ) ,0,\ldots ,0\right ) $ . We have not been able to provide primitive conditions for this to hold when K can take negative values and so instead impose the positivity constraint on K.

If the objective function $\theta \mapsto \ell _{n,t}\left ( \theta \right ) $ is concave and $\Theta $ is convex, we can replace Assumption 3 with the following pointwise versions: For any $\theta \in \Theta $ , $\ell _{n,t}\left ( \theta \right ) $ is locally stationary and $\mathbb {E}\left [ |\ell _{t}^{\ast }\left ( \theta |u\right ) |\right ] <\infty $ (see Theorem 2.7 in Newey and McFadden, Reference Newey and McFadden1994). Under the above assumptions, the following consistency result holds.

Note that the above theorem only shows the consistency of $\hat {\theta }\left ( u\right ) $ , and so, at this stage, we cannot make any statements regarding $ \hat {\theta }^{\left ( i\right ) }\left ( u\right ) $ , $i=1,\ldots ,m$ . This is similar to other results for nonlinear extremum estimators that converge with different rates (see, e.g., Theorem 9 in Han and Kristensen, Reference Han and Kristensen2014, where a global consistency result is only provided for the component with the fastest rate).

However, under additional regularity conditions on the quasi-likelihood function, we can provide a more precise analysis of the estimators, including local consistency of $\hat {\theta }^{\left ( k\right ) }\left ( u\right ) $ , $1\leq k\leq m$ . With $s_{n,t}\left ( \theta \right ) =\partial \ell _{n,t}\left ( \theta \right ) /\left ( \partial \theta \right ) \in \mathbb { R}^{d_{\theta }}$ and $h_{n,t}\left ( \theta \right ) =\partial ^{2}\ell _{n,t}\left ( \theta \right ) /(\partial \theta \partial \theta ^{^{\prime }})\in \mathbb {R}^{d_{\theta }\times d_{\theta }}$ , $D_{n,t}\left ( u\right ) =D_{m,b}\left ( t/n-u\right ) $ and $K_{n,t}\left ( u\right ) =K_{b}(t/n-u)$ , the score and Hessian of $Q_{n}\left ( \alpha |u\right ) $ are given by

$$ \begin{align*} S_{n}\left( \alpha |u\right) & =\frac{\partial Q_{n}\left( \alpha |u\right) }{\partial \alpha }=\frac{1}{n}\sum_{t=1}^{n}K_{n,t}\left( u\right) D_{n,t}\left( u\right) ^{\prime }s_{n,t}\left( D_{n,t}\left( u\right) \alpha \right) , \\ H_{n}\left( \alpha |u\right) & =\frac{\partial ^{2}Q_{n}\left( \alpha |u\right) }{\partial \alpha \partial \alpha ^{\prime }}=\frac{1}{n} \sum_{t=1}^{n}K_{n,t}\left( u\right) D_{n,t}\left( u\right) ^{\prime }h_{n,t}\left( D_{n,t}\left( u\right) \alpha \right) D_{n,t}\left( u\right). \end{align*} $$

It is easily checked that $\alpha _{0}:=U_{n}\beta _{0}$ , where $\beta _{0}=(\theta \left ( u\right ) ^{\prime },\theta ^{\left ( 1\right ) }\left ( u\right ) ^{\prime },\ldots ,\theta ^{\left ( m\right ) }\left ( u\right ) ^{\prime })^{\prime }$ , belongs to the interior of $\mathcal {A}$ for all n large enough due to Assumption 4(ii) below in conjunction with Assumption 2. Due to the consistency result, so will $\hat {\alpha }$ w.p.a. 1. Thus, $\hat {\alpha }$ will satisfy the first-order condition of (7) which combined with the mean-value theorem yields

(9) $$ \begin{align} 0=S_{n}\left( \hat{\alpha}|u\right) =S_{n}\left( \alpha _{0}|u\right) +H_{n}\left( \bar{\alpha}|u\right) \left( \hat{\alpha}-\alpha _{0}\right) =S_{n}\left( \alpha _{0}|u\right) +H_{n}\left( \bar{\alpha}|u\right) U_{n}( \hat{\beta}-\beta _{0}), \end{align} $$

where $\bar {\alpha }$ is situated on the line segment connecting $\hat {\alpha } $ and $\alpha _{0}$ . We then decompose the score function into a bias and variance component, $S_{n}\left ( \alpha _{0}|u\right ) =B_{n}\left ( u\right ) +S_{n}\left ( u\right ) $ , where

(10) $$ \begin{align} B_{n}\left( u\right) &=\frac{1}{n}\sum_{t=1}^{n}K_{n,t}\left( u\right) D_{n,t}\left( u\right) ^{\prime }b_{n,t},\;\ \ b_{n,t}=s_{n,t}\left( \theta _{u}^{\ast }\left( t/n\right) \right) -s_{n,t}\left( \theta \left( t/n\right) \right) \end{align} $$

(11) $$ \begin{align} S_{n}\left( u\right) &=\frac{1}{n}\sum_{t=1}^{n}K_{n,t}\left( u\right) D_{n,t}\left( u\right) ^{\prime }s_{n,t},\;\ \ s_{n,t}=s_{n,t}\left( \theta \left( t/n\right) \right) \end{align} $$

and $\theta _{u}^{\ast }\left ( t/n\right ) $ was defined in eq. (1). This decomposition is different from the one usually employed in the analysis of kernel estimators of time-varying coefficients, where $ s_{n,t}$ is replaced by the stationary version of the score function evaluated at $\theta \left ( u\right ) $ , $s_{t}^{\ast }\left ( \theta \left ( u\right ) |u\right ) $ (see, e.g., Dahlhaus and Subba Rao, Reference Dahlhaus and Subba Rao2006; Dahlhaus et al., Reference Dahlhaus, Richter and Wu2019). This “usual” choice has the consequence that the corresponding bias term generally involves the time derivative process of the score function and so the resulting analysis tends to impose stronger regularity conditions on the model being estimated. By instead centering the analysis around $s_{n,t}$ , we can obtain the leading term of the bias $ B_{n}\left ( u\right ) $ through a standard Taylor expansion w.r.t. $\theta $ :

(12) $$ \begin{align} b_{n,t}\cong h_{n,t}\left( \theta _{u}^{\ast }\left( t/n\right) \right) \left\{ \theta _{u}^{\ast }\left( t/n\right) -\theta \left( t/n\right) \right\} \cong -h_{n,t}\left( \theta \left( u\right) \right) \frac{\theta ^{\left( m+1\right) }\left( u\right) }{\left( m+1\right) !}\left\{ t/n-u\right\} ^{m+1}. \end{align} $$

Thus, our approach allows for a simpler derivation of the leading bias and variance terms under the following weak regularity conditions.

Similar to Assumption 2, Assumption 4 contains standard regularity conditions used in the analysis of regular parametric estimators on the form $\tilde {\theta }\left ( u\right ) =\arg \max _{\theta \in \Theta }\sum _{t=1}^{n}\ell _{t}^{\ast }\left ( \theta |u\right ) $ . At the same time, Assumption 4(iii) is non-standard compared to the existing literature (as discussed above) and allows us to apply a novel martingale central limit theorem for locally stationary sequences to $S_{n}\left ( u\right ) $ :

(13) $$ \begin{align} \sqrt{nb}S_{n}\left( u\right) \rightarrow ^{d}N\left( 0,\mathbb{K} _{2}\otimes \Omega \left( u\right) \right) ,\;\ \ \Omega \left( u\right) = \mathbb{E}\left[ s_{t}^{\ast }\left( \theta \left( u\right) |u\right) s_{t}^{\ast }\left( \theta \left( u\right) |u\right) ^{\prime }\right] \end{align} $$

(see Lemma A(iii) in Appendix A.1). This result can be seen as a generalization of the standard CLT for stationary and ergodic MGDs that allows for locally stationary processes. The MGD assumption amounts to assuming that the time-varying model is correctly specified and has to be verified on a case-by-case basis (see Section 4.2 for examples of this).

Finally, Assumption 5 together with the expansion in eq. ( 12) is used to derive the limits of $B_{n}\left ( u\right ) $ and $H_{n}\left ( \bar {\alpha }|u\right ) $ :

(14) $$ \begin{align} H_{n}\left( \bar{\alpha}|u\right) \rightarrow ^{p}\mathbb{K}_{1}\otimes H\left( u\right) ,\;\ \ B_{n}\left( u\right) =-b^{m+1}\mu _{1}\otimes H\left( u\right) \frac{\theta ^{\left( m+1\right) }\left( u\right) }{\left( m+1\right) !}+o_{P}\left( b^{m+1}\right), \end{align} $$

where $\mu _{i}=\int _{\mathbb {R}}K\left ( v\right ) v^{m+i}D_{m}\left ( v\right ) dv$ and $\mathbb {K}_{i}=\int _{\mathbb {R}}K^{i}\left ( v\right ) D_{m}\left ( v\right ) D_{m}\left ( v\right ) ^{\prime }dv$ , $i\geq 1$ . Combining (9), (13), and (14), we obtain the following theorem.

Theorem 2. Suppose that Assumptions 1–5 hold. Then, as $b\rightarrow 0$ and $nb\rightarrow \infty $ ,

$$ \begin{align*} \sqrt{nb}U_{n}\left\{ \hat{\beta}-\beta _{0}-R_{n}Bias\left( u\right) \right\} \rightarrow ^{d}N\left( 0,\mathbb{K}_{1}^{-1}\mathbb{K}_{2}\mathbb{K }_{1}^{-1}\otimes V(u)\right), \end{align*} $$

where $R_{n}=diag\left \{ b^{m+1},b^{m},\ldots ,b\right \} \otimes I_{d_{\theta }}$ , $V(u)=H\left ( u\right ) ^{-1}\Omega \left ( u\right ) H\left ( u\right ) ^{-1}$ , and $Bias\left ( u\right ) =\mathbb {K}_{1}^{-1}\mu _{1}\otimes \frac {\theta ^{\left ( m+1\right ) }\left ( u\right ) }{\left ( m+1\right ) !}$ .

In particular, for $i=0,1,\ldots ,m$ ,

(15) $$ \begin{align} \sqrt{nb^{2i+1}}\left\{ \hat{\theta}^{\left( i\right) }\left( u\right) -\theta ^{\left( i\right) }\left( u\right) -b^{m+1-i}Bias_{i}\left( u\right) \right\} \rightarrow ^{d}N\left( 0,\kappa _{2,i}V\left( u\right) \right), \end{align} $$

where $Bias_{i}\left ( u\right ) =\kappa _{1,i}\frac {\theta ^{\left ( m+1\right ) }\left ( u\right ) }{\left ( m+1\right ) !}$ and $\kappa _{1,i}$ and $ \kappa _{2,i}$ denote the ith element of $\mathbb {K}_{1}^{-1}\mu _{1}$ and $\left ( i,i\right ) $ th element of $\mathbb {K}_{1}^{-1}\mathbb {K}_{2}\mathbb {K }_{1}^{-1}$ , respectively.

Similar to existing results for local polynomial estimators in a cross-sectional setting, the leading bias term in (15) only depends on $\theta ^{\left ( m+1\right ) }\left ( u\right ) $ and so the estimators adapt to the curvature of $\theta \left ( u\right ) $ . The asymptotic variance in Theorem 2 can be estimated using plug-in methods: It follows from the proof of Theorem 2 that $ H_{n}\left ( \hat {\alpha }|u\right ) \rightarrow ^{p}\mathbb {K}_{1}\otimes H\left ( u\right ) $ and

$$ \begin{align*} \hat{W}\left( u\right) &=\frac{1}{n}\sum_{t=1}^{n}K_{n,t}^{2}\left( u\right) D_{n,t}\left( u\right) ^{\prime }s_{n,t}\left( D_{n,t}\left( u\right) \hat{ \alpha}\right) s_{n,t}\left( D_{n,t}\left( u\right) \hat{\alpha}\right) ^{\prime }D_{n,t}\left( u\right)\\&\quad \rightarrow ^{p}\mathbb{K}_{2}\otimes \Omega \left( u\right). \end{align*} $$

Compared to most existing asymptotic results for the local constant estimator, such as Dahlhaus et al. (Reference Dahlhaus, Richter and Wu2019), the above result with $m\geq 1$ imposes much weaker restrictions on the bandwidth. In particular, standard bandwidth selection rules can be employed here but not under most of the existing theories since their conditions require undersmoothing (i.e., $ b\rightarrow 0$ at a faster rate than the optimal one). This is due to the fact that these theories do not provide a complete characterization of the leading bias term. The few papers that do characterize the leading bias term, such as Dahlhaus and Subba Rao (Reference Dahlhaus and Subba Rao2006), require the so-called time derivatives of the stationary score function to exist and be well-behaved since these enter their bias expressions. Our conditions and results, on the other hand, do not require these and are analogous to the ones found in the literature on local polynomial likelihood estimators (see, e.g., Theorem 1b of Fan et al., Reference Fan, Heckman and Wand1995).

Equation (15) holds for any value of $m\geq 0$ and $ i=0,\ldots ,m$ . However, if K is symmetric, then $\kappa _{1,i}=0$ when $m-i$ is even. In particular, for the local constant estimator ( $m=i=0$ ), Theorem 2 only informs us that the bias component of $\hat {\theta } \left ( u\right ) $ is $o_{p}\left ( b\right ) $ which is not a sharp rate. To obtain the leading bias term in the cases where $m-i$ is even, a higher-order expansion of $b_{n,t}$ in eq. (10) is necessary. This expansion requires additional assumptions involving aforementioned time derivatives and standard derivatives w.r.t. $\theta $ of $h_{t}^{\ast }\left ( \theta \left ( u\right ) |u\right ) $ . To present these, we need the following additional concept.

Definition 3. A stationary process $W_{t}^{\ast }\left ( \theta |u\right ) $ is said to be $ L_{p}$ -differentiable w.r.t. u if there exists a stationary and ergodic process $\partial _{u}W_{t}^{\ast }\left ( \theta |u\right ) $ with $ \mathbb {E}\left [ \left \Vert \partial _{u}W_{t}^{\ast }\left ( \theta |u\right ) \right \Vert ^{p}\right ] <\infty $ such that

$$ \begin{align*} \mathbb{E}\left[ \left\Vert W_{t}^{\ast }\left( \theta |u+\Delta \right) -W_{t}^{\ast }\left( \theta |u\right) -\partial _{u}W_{t}^{\ast }\left( \theta |u\right) \Delta \right\Vert ^{p}\right] ^{1/p}=o\left( \Delta \right) ,\;\Delta \rightarrow 0. \end{align*} $$

Our definition of time differentiability is slightly weaker compared to the one found in Dahlhaus et al. (Reference Dahlhaus, Richter and Wu2019) and other papers where differentiability w.r.t. u has to hold almost surely. With this definition in hand, we are ready to introduce the following additional regularity conditions in order to derive the leading bias term when $m-i$ is even.

The time-derivative $\partial _{u}h_{t}^{\ast }\left ( \theta |u\right ) $ will generally involve time derivatives of the underlying stationary approximation of data: If $h_{t}^{\ast }\left ( \theta |u\right ) =h\left ( \mathcal {Z}_{t}^{\ast }\left ( u\right ) ;\theta \right ) $ for some function $ h $ which is differentiable w.r.t. $\mathcal {Z}_{t}^{\ast }\left ( u\right ) $ , then it takes the form $\partial _{u}h_{t}^{\ast }\left ( \theta |u\right ) =\sum _{i=0}^{\infty }\partial h\left ( z_{0},z_{1},z_{2},\dots ;\theta \right ) /\left ( \partial z_{i}\right ) |_{z=\mathcal {Z}_{t}^{\ast }\left ( u\right ) }\times \partial _{u}Z_{t-i}^{\ast }\left ( u\right ) $ , where $\partial _{u}Z_{i,t}^{\ast }\left ( u\right ) $ is the time derivative of $Z_{t}^{\ast }\left ( u\right ) $ . The short memory condition imposed in Assumption 8 is used to control the variance component of the first-order bias term derived in Theorem 2. In Section 4, we use the concept of $\tau $ —weak dependence (Doukhan and Wintenberger, Reference Doukhan and Wintenberger2008) to verify Assumption 8.

Under the above additional assumptions, we obtain the following higher-order expansion of the bias component.

Theorem 3. Suppose Assumptions 1–8 hold and $\theta \left ( \cdot \right ) $ is $m+2$ times continuously differentiable. Then, as $b\rightarrow 0$ and $nb\rightarrow \infty $ , the bias $B_{n}\left ( u\right ) $ defined in (10) satisfies, with r given in Assumption 6,

(16) $$ \begin{align} B_{n}\left( u\right) &=-b^{m+1}\left[ \mu _{1}\otimes H\left( u\right) \frac{\theta ^{\left( m+1\right) }\left( u\right) }{\left( m+1\right) !} +o_{P}\left( b\right) \right] -b^{m+2}\left[ \mu _{2}\otimes B_{1}\left( u\right) +o_{p}\left( 1\right) \right] \\ &\quad-b^{2m+2}\left[ \mu _{2}\otimes B_{2}\left( u\right) +o_{p}\left( 1\right) \right] +O_{P}\left( 1/n^{\min \left\{ 1,r\right\} }\right) +o_{p}\left( \frac{1}{\sqrt{nb}}\right) , \nonumber \end{align} $$

where, with $\partial _{u}H\left ( u\right ) =\mathbb {E}\left [ \partial _{u}h_{t}^{\ast }\left ( \theta |u\right ) \right ] _{\theta =\theta \left ( u\right ) }$ and $\partial _{\theta _{i}}H\left ( u\right ) =\mathbb {E}\left [ \partial h_{t}^{\ast }\left ( \theta |u\right ) /\left ( \partial \theta _{i}\right ) \right ] _{\theta =\theta \left ( u\right ) }$ ,

$$ \begin{align*} B_{1}\left( u\right) & =H\left( u\right) \frac{\theta ^{\left( m+2\right) }\left( u\right) }{\left( m+2\right) !}+\left( \partial _{u}H\left( u\right) +\sum_{i=1}^{d_{\theta }}\theta _{i}^{\left( 1\right) }\left( u\right) \partial _{\theta _{i}}H\left( u\right) \right) \frac{\theta ^{\left( m+1\right) }\left( u\right) }{\left( m+1\right) !}, \\ B_{2}\left( u\right) & =\frac{-1}{2\left\{ \left( m+1\right) !\right\} ^{2}} \sum_{i=1}^{d_{\theta }}\theta _{i}^{\left( m+1\right) }\left( u\right) \partial _{\theta _{i}}H\left( u\right) \theta ^{\left( m+1\right) }\left( u\right). \end{align*} $$

As a special case, we obtain the following result for the local constant estimator.

Corollary 1. Under the assumptions of Theorem 3 with $ m=0$ together with $\int _{\mathbb {R}}K\left ( v\right ) vdv=0$ , the local constant estimator satisfies, as $b\rightarrow 0$ , $nb^{3}\rightarrow \infty , $ and $n^{\min \left \{ r,1\right \} }b\rightarrow \infty $ ,

$$ \begin{align*} \sqrt{nb}\left\{ \hat{\theta}\left( u\right) -\theta \left( u\right) -b^{2}\mu _{2}Bias_{0}\left( u\right) \right\} \rightarrow ^{d}N\left( 0,\kappa _{2,0}V\left( u\right) \right), \end{align*} $$

where $Bias_{0}\left ( u\right ) =H^{-1}\left ( u\right ) \left [ B_{1}(u)+B_{2}\left ( u\right ) \right ] $ and $B_{1}(u)$ and $B_{2}(u)$ are given in Theorem 3.

Two equivalent representations of $B_{1}(u)+B_{2}\left ( u\right ) $ are, with $s_{t}^{\ast }\left ( \theta |u\right ) $ denoting the stationary approximation of $s_{n,t}\left ( \theta \right ) $ ,

$$ \begin{align*} B_{1}(u)+B_{2}\left( u\right) &=\frac{1}{2}\frac{\partial ^{2}}{\partial v^{2}}\mathbb{E}\left[ s_{t}^{\ast }\left( \theta \left( v\right) |u\right) \right] _{v=u}+\frac{\partial ^{2}}{\partial u\partial v}\mathbb{E}\left[ s_{t}^{\ast }\left( \theta \left( v\right) |u\right) \right] _{v=u} \\ &=-\frac{1}{2}\frac{\partial ^{2}}{\partial u^{2}}\mathbb{E}\left[ s_{t}^{\ast }\left( \theta |u\right) \right] _{\theta =\theta \left( u\right) }, \end{align*} $$

where the second representation is only well-defined if $s_{t}^{\ast }\left ( \theta |u\right ) $ is twice $L_{1}$ -differentiable w.r.t. u.

To our knowledge, this is the first complete characterization of the leading bias term of local constant estimators in general time-varying parameter models. The final characterization of the bias, $B_{1}(u)+B_{2}\left ( u\right ) =-\frac {1}{2}\partial _{u}^{2}\mathbb {E}\left [ s_{t}^{\ast }\left ( \theta \left ( v\right ) |u\right ) \right ] _{v=u}$ , corresponds to the one obtained in Dahlhaus and Subba Rao (Reference Dahlhaus and Subba Rao2006) for the time-varying ARCH model. This characterization, however, requires $s_{t}^{\ast }\left ( \theta |u\right ) $ to be twice differentiable w.r.t. u, while our characterization only requires $h_{t}^{\ast }\left ( \theta |u\right ) $ to be once differentiable w.r.t. u.

Comparing Theorems 2 and 3, we see that the local linear and local constant estimators share convergence rate and asymptotic variance, but that the latter suffers from additional biases. This is consistent with the theory found for local constant and local linear estimators in a cross-sectional setting (see, e.g., Fan et al., Reference Fan, Heckman and Wand1995).

3.2 Behavior at Boundary

We have already seen that the local linear estimator has smaller biases than the local constant one in the interior of its domain, $u\in \left ( 0,1\right ) $ . Another well-known advantage of the local linear estimators in a cross-sectional setting is that they exhibit automatic boundary carpentering. This property also holds in our setting where the boundaries are $u=0$ and $u=1$ . Since the results for $u=1$ are similar, we here only analyze the properties of the local constant ( $m=0$ ) and the local linear ( $ m=1$ ) estimators at $u=cb$ for some constant $c>0$ . Combining the intermediate bias–variance analysis carried out in the proofs of Theorem 2 and 3 with the arguments of Fan et al. (Reference Fan, Heckman and Wand1995), we find that the two estimators remain asymptotically normally distributed but their asymptotic biases and variances take different forms.

Corollary 2. Let $\hat {\theta }_{0}\left ( u\right ) $ and $\hat {\theta } _{1}\left ( u\right ) $ be the local constant and local linear estimators, respectively, of $\theta \left ( u\right ) $ . Under the assumptions of Theorem 3 with $m=1$ together with $\mu _{1}=\int _{\mathbb {R}}K\left ( v\right ) vdv=0$ , as $b\rightarrow 0$ , $nb^{3}\rightarrow \infty , $ and $ n^{\min \left \{ r,1\right \} }b\rightarrow \infty $ , and for $m\in \{0,1\}$ ,

$$ \begin{align*} \sqrt{nb}\left( \hat{\theta}_{m}\left( cb\right) -\theta \left( cb\right) -b^{m+1}Bias_{m}\right) \overset{d}{\rightarrow }N\left( 0,a_{m}V\left( 0_{+}\right) \right), \end{align*} $$

where $V\left ( 0_{+}\right ) =\lim _{u\downarrow 0}V\left ( u\right ) $ and

$$ \begin{align*} Bias_{0}&=\left( \kappa _{1,0}^{c}\right) ^{-1}\kappa _{1,1}^{c}\theta ^{\left( 1\right) }\left( 0_{+}\right) ;\quad Bias_{1}=\frac{1}{2}\frac{ \left( \kappa _{1,2}^{c}\right) ^{2}-\kappa _{1,1}^{c}\kappa _{1,3}^{c}}{ \kappa _{1,0}^{c}\kappa _{1,2}^{c}-\left( \kappa _{1,1}^{c}\right) ^{2}} \theta ^{\left( 2\right) }\left( 0_{+}\right); \\ a_{0}&=\kappa _{2,0}^{c}/\left( \kappa _{1,0}^{c}\right) ^{2};\\ a_{1}&= \left[ \left( \kappa _{1,2}^{c}\right) ^{2}\kappa _{2,0}^{c}-2\kappa _{1,1}^{c}\kappa _{1,2}^{c}\kappa _{2,1}^{c}+\left( \kappa _{1,1}\right) ^{2}\kappa _{2,2}^{c}\right] /\left[ \kappa _{1,0}^{c}\kappa _{1,2}^{c}-\left( \kappa _{1,1}^{c}\right) ^{2}\right] ^{2}. \end{align*} $$

We refer to Fan et al. (Reference Fan, Heckman and Wand1995) for the precise expressions of the constants $\kappa _{i,j}^{c}$ , $i=1,2$ and $j=0,1,2$ . At the boundary, both the biases and variances of the local constant and local linear estimators are now different. While the difference between two asymptotic variances is a constant scale, compare $a_{0}$ and $a_{1}$ above, the biases are now of a different order: The local linear estimator still enjoys a bias of order $ O\left ( b^{2}\right ) $ while the bias of the local constant one blows up and becomes of order $O\left ( b\right ) $ . Thus, the local constant estimator will generally suffer from significantly larger biases at the boundary compared to the local linear one.

4 APPLICATIONS

In this section, we illustrate how our general results can be verified for a class of Markov models with exogenous covariates and provide concrete examples within this framework. Application to infinite-memory models, as considered in Bardet et al. (Reference Bardet, Doukhan and Wintenberger2022), is discussed in Section 7 as a direction for future work.

By inspection of our Assumptions 1–5, we observe that, for a given parametric model and estimator, most of the assumptions are easily verifiable using standard arguments known from the literature on regular parametric estimators. The only ones that are non-standard are Assumptions 3 and 5, which require the researcher to show that $\ell _{n,t}\left ( \theta \right ) $ and its first two derivatives are ULS. Similarly, Assumption 6 is also a ULS requirement, while Assumption 8 and/or 10 will hold if $h_{t}^{\ast }\left ( \theta \left ( u\right ) |u\right ) $ is weakly dependent.

The following lemma provides sufficient conditions for a given transformation of a time series to be ULS if the underlying time series is ULS. It also shows that if the stationary version is $\tau $ -weakly dependent (see Doukhan and Wintenberger, Reference Doukhan and Wintenberger2008 for the definition), then so is the transformation.

Next, we here provide more primitive conditions for Assumption 9 that can be applied to discrete-valued random variables.

A sufficient condition for $\int \left \vert f\left ( z\right ) \partial _{u}p\left ( zu\right ) \right \vert d\mu \left ( z\right ) <\infty $ is $\mathbb { E} [ \left \vert f\left ( Z_{t}^{\ast }\left (u\right ) \right )\right \vert \left \vert s\left ( Z_{t}^{\ast }\left (u\right )\right ) \right \vert ] <\infty $ , where $s = \partial _{u}p/p$ is the score function. For example, if $\left \vert s\left ( z;u\right ) \right \vert \leq C\left ( 1+\left \Vert z\right \Vert ^{\delta }\right ) $ , $\delta \geq 0$ , then the result will hold under $E\left [ \left \Vert Z_{t}^{\ast }\left ( \theta |u\right ) \right \Vert ^{\delta }\left \Vert f\left ( Z_{t}^{\ast }\left ( \theta |u\right ) ;u\right ) \right \Vert \right ] <\infty $ . For Markov models, the conditions of Lemma 2 can be verified by importing results from Truquet (Reference Truquet2020) and Vazquez-Abad and Kushner (Reference Vazquez-Abad and Kushner1992); see proof of Corollary 6 for an example of this.

We summarize our findings in the following corollary.

If the model and estimator of interest are regular in the sense that its stationary version satisfies the above assumptions, then all that remains to be shown is that $Z_{n,t}$ is locally stationary with $Z_{t}^{\ast }\left ( u\right ) $ being weakly dependent. The next section provides primitive conditions for this to hold for Markov models with exogenous covariates, while the second section revisits the examples of Section 2 and provides primitive conditions under which our main results apply to these.

4.1 Markov Models with Exogenous Covariates

Consider the following class of q-Markov models with exogenous covariates:

(18) $$ \begin{align} Y_{n,t}=G\left( Y_{n,t-1},\ldots,Y_{n,t-q},X_{n,t-1},\varepsilon _{t};\theta \left( t/n\right) \right) , \end{align} $$

(19) $$ \begin{align} X_{n,t}=H\left( X_{n,t-1},\ldots,X_{n,t-q},\eta _{t},t/n\right). \end{align} $$

We here restrict $X_{n,t}$ to be a Markov process. We could allow for $ X_{n,t}$ to exhibit richer dynamics, for example, allow for $X_{n,t}$ to depend on lags of $Y_{n,t}$ . However, this would lead to more complicated assumptions and so we here maintain (19) for simplicity. We impose the following assumptions on (18) and (19).

Assumption 11. (i) $\left ( \varepsilon _{t},\eta _{t}\right ) $ is i.i.d. over time; (ii) for some $\rho _{x}<1$ , $p\geq 1,$ and $x_{0}\in \mathcal {X}^{q}$ , $\mathbb {E}\left [ \left \Vert H\left ( x;\eta _{t},u\right ) -H\left ( \tilde {x};\eta _{t},u\right ) \right \Vert ^{p}\right ] \leq \rho _{x}\left \Vert x-\tilde {x}\right \Vert ^{p}$ and $\mathbb {E}\left [ \left \Vert H\left ( x_{0};\eta _{t},u\right ) \right \Vert ^{p}\right ] <\infty $ for all $ x,x^{\prime }\in \mathcal {X}^{q}$ and $u\in \left [ 0,1\right ] $ ; (iii) for some $\left ( x_{0},y_{0}\right ) \in \mathcal {X}^{q}\times \mathcal {Y}^{q}$ , $ L<\infty $ and $\rho _{y}<1$ , $\sup _{\theta \in \Theta }\mathbb {E}\left [ \lVert G\left ( y_{0},x_{0},\varepsilon _{t};\theta \right ) \rVert ^{p}\right ] <\infty $ and

$$ \begin{align*} \mathbb{E}\left[ \lVert G\left( y,x,\varepsilon _{t};\theta \right) -G\left( y^{\prime },x^{\prime },\varepsilon _{t};\theta \right) \rVert ^{p}\right] ^{1/p}\leq L\lVert x-x^{\prime }\rVert +\rho _{y}\lVert y-y^{\prime }\rVert \end{align*} $$

for all $\left ( x,y\right ) ,\left ( x^{\prime },y^{\prime }\right ) \in \mathcal {X}^{q}\times \mathcal {Y}^{q}$ ; (iv) for some $\tilde {p}\geq 1$ , $ r>0, $ and $\delta \geq 0$ and all $\theta ,\theta ^{\prime }\in \Theta $ ,

$$ \begin{align*} \mathbb{E}\left[ \lVert G\left( y,x,\varepsilon _{t};\theta \right) -G\left( y,x,\varepsilon _{t};\theta ^{\prime }\right) \rVert ^{\tilde{p}}\right] ^{1/ \tilde{p}}\leq C\left( 1+\lVert x\rVert ^{\delta }+\lVert y\rVert ^{\delta }\right) \lVert \theta -\theta ^{\prime }||^{r}; \end{align*} $$

$$ \begin{align*} \mathbb{E}\left[ \lVert H\left( x,\eta _{t};u\right) -H\left( x,\eta _{t};v\right) \rVert ^{\tilde{p}}\right] ^{1/\tilde{p}}\leq C\left( 1+\lVert x\rVert ^{\delta }\right) \left\vert u-v\right\vert ^{r}; \end{align*} $$

(v) $\mathbb {E}\left [ \lVert Y_{n,0}\rVert ^{\tilde {p}}\right ] <\infty $ and $\mathbb {E}\left [ \lVert X_{n,0}\rVert ^{\tilde {p}}\right ] <\infty $ .

This assumption is similar to the one found in Dahlhaus et al. (Reference Dahlhaus, Richter and Wu2019). However, in contrast to this article, we do not require the process to be initialized at the stationary distribution, $(Y_{n,0},X_{n,0})=(Y_{0}^{* }\left ( u\right ),X_{0}^{*}\left (u\right ))$ , and instead allow it to be initialized at any given value or distribution.

Lemma 3. Under Assumption 11(i)–(iii), for any $u\in \left [ 0,1\right ] $ , there exists a stationary and $\tau $ -weakly dependent solution $\left \{ Y_{t}^{\ast }\left ( u\right ) ,X_{t}^{\ast }\left ( u\right ) \right \} $ to

(20) $$ \begin{align} Y_{t}^{\ast }\left( u\right) & = G\left( Y_{t-1}^{\ast }\left( u\right) ,X_{t-1}^{\ast }\left( u\right) ,\varepsilon _{t};\theta \left( u\right) \right) , \end{align} $$

(21) $$ \begin{align} X_{t}^{\ast }\left( u\right) & = H\left( X_{t-1}^{\ast }\left( u\right) ,\eta _{t},u\right) , \end{align} $$

with $\tau _{Y,X}\left ( k\right ) \leq O\left ( \rho ^{k}\right ) $ . If furthermore Assumption 11(iv) and (v) hold, $\sup _{u\in \left [ 0,1 \right ] }\mathbb {E}\left [ \lVert X_{t}^{\ast }\left ( u\right ) \rVert ^{ \tilde {p}\delta }\right ] $ , $\sup _{u\in \left [ 0,1\right ] }\mathbb {E}\left [ \lVert Y_{t}^{\ast }\left ( u\right ) \rVert ^{\tilde {p}\delta }\right ] <\infty $ and $u\mapsto \theta \left ( u\right ) $ and $u\mapsto H\left ( x,\eta _{t},u\right ) $ is continuously differentiable, then $\left ( Y_{n,t},X_{n,t}\right ) $ is LS $\left ( \tilde {p},r\right ) $ with $\sup _{n,t} \mathbb {E}\left [ \lVert Y_{n,t}\rVert ^{\tilde {p}}\right ] <\infty $ and $ \sup _{n,t}\mathbb {E}\left [ \lVert X_{n,t}\rVert ^{\tilde {p}}\right ] <\infty $ for any given initial values $\left ( Y_{n,0},X_{n,0}\right ) $ with $\mathbb {E }\left [ \lVert Y_{n,0}\rVert ^{p}\right ] <\infty $ and $\mathbb {E}\left [ \lVert X_{n,0}\rVert ^{p}\right ] <\infty $ .

Combining Corollary 3 with Lemma 3, we obtain a set of easily verifiable primitive conditions for local M-estimators of time-varying parameters in Markov models to satisfy Theorems 2 and 4 (see Section 4.2 for examples of the verification procedure).

4.2 Specific Examples

We here apply Corollary 3 and Lemma 3 to Examples 1–3. Note here that the first and third examples cannot be analyzed using some of the existing theories since these rely on differentiability of the DGP. Moreover, none of the existing theories allow for covariates to be included in the model. As such, the results below are new to the literature.

Throughout this section, the following assumption is maintained where $\theta \left ( u\right ) $ and $\Theta $ are specified in each of the following examples.

Assumption 12. (i) The kernel K satisfies Assumption 1; (ii) $\theta \left ( u\right ) \in \text {Int}\left ( \Theta \right ) $ and $\theta \left ( \cdot \right ) $ is twice continuously differentiable; (iii) the additional predictors $X_{n,t}$ of the model solve (19), where $H\left ( x,\eta _{\tau };u\right ) $ satisfies the conditions in Assumption 11 with $p=\tilde {p}=2$ ; and (iv) $\varepsilon _{t}$ is i.i.d.

For the time-varying TAR-X model in Example 1, we obtain the following result.

Corollary 4. Let $\Theta =\mathbb {R}^{2q+d_{X}+1}$ and suppose that Assumption 12 holds with $\mathbb {E}\left [ \varepsilon _{t}^{2}\right ] <\infty $ and

(22) $$ \begin{align} \sup_{u\in \left[ 0,1\right] }\sum_{i=1}^{q}\max \left\{ \left\vert \alpha _{1,i}\left( u\right) \right\vert ,\left\vert \alpha _{2,i}\left( u\right) \right\vert \right\} <1. \end{align} $$

Then, the results of Theorem 2 apply to the local linear estimators of the tv-TAR $(1)$ model (4) with $H\left ( u\right ) =\mathbb {E}\left [ \tilde {X}_{t}^{\ast }\left ( u\right ) \tilde {X} _{t}^{\ast }\left ( u\right ) ^{\prime }\right ] $ , $\Omega \left ( u\right ) = \mathbb {E}\left [ \varepsilon _{t}^{2}\right ] H\left ( u\right )$ , and

$$ \begin{align*} \tilde{X}_{t}^{\ast }\left( u\right) =\left( Y_{t-1}^{\ast +}\left( u\right) ,\dots,Y_{t-q}^{\ast +}\left( u\right) ,Y_{t-1}^{\ast -}\left( u\right) ,\dots,Y_{t-q}^{\ast -}\left( u\right) ,X_{t-1}^{\ast }\left( u\right) \right). \end{align*} $$

If, in addition, $\mathbb {E}\left [ \varepsilon _{t}^{4}\right ] <\infty $ , $ \varepsilon _{t}$ has a differentiable density and $H\left ( x,\eta _{t},u\right ) $ satisfies Assumption 4.1(L3) and (L4) in Dahlhaus et al. (Reference Dahlhaus, Richter and Wu2019), then the results of Theorem 3 hold for the local constant estimators with $\partial _{\theta }H\left ( u\right ) =0$ and $\partial _{u}H\left ( u\right ) =\partial H\left ( \theta |u\right ) /\left ( \partial u\right ) |_{\theta =\theta \left ( u\right ) }$ .

Note here that we do not need to restrict $\Theta $ to be compact since here $\ell _{n,t}\left ( \theta \right ) $ is concave. Equation (22) ensures that the model indeed has a locally stationary solution and is $\tau $ -weakly dependent. The additional restrictions imposed in the second part of the corollary are used to show that the time derivative of $\left ( Y_{t}^{\ast }\left ( u\right ) ,X_{t}^{\ast }\left ( u\right ) \right ) $ exists so that Assumption 7 holds.

Next, consider the local Gaussian QMLE of the tv-ARCH-X model given in Example 2. Under eq. (23) below, there exists a locally stationary solution to the model which takes the form

$$ \begin{align*} Y_{t}^{\ast }(u)=\lambda _{t}^{\ast }(u)\varepsilon _{t}^{2},\;\ \ \lambda _{t}^{\ast }(u)=\theta (u)^{\prime }\tilde{X}_{t}^{\ast }(u), \end{align*} $$

where $\tilde {X}_{t}^{\ast }(u)=\left ( 1,Y_{t-1}^{\ast }(u),\ldots ,Y_{t-q}^{\ast }(u),X_{t-1}^{\ast }\left ( u\right ) ^{\prime }\right ) ^{\prime }$ .

Corollary 5. Assume that Assumption 12 holds, $\Theta \subseteq \mathbb {R}_{+}^{1+q+d_{X}}$ is compact with $\omega \geq \underline {\omega }>0$ for all $\theta \in \Theta $ , $X_{n,t-1}\in \mathbb {R} _{+}^{d_{X}}$ , $\mathbb {E}\left [ \varepsilon _{t}^{4}\right ] <\infty $ , $ \inf _{u\in \left [ 0,1\right ] }\omega \left ( u\right )>0,$ and

(23) $$ \begin{align} \sup_{u\in \left[ 0,1\right] }\sum_{i=1}^{q}\alpha _{i}\left( u\right) <1. \end{align} $$

Then, the results of Theorem 2 apply to the local linear estimators of the tv-ARCH-X model (5) with

$$ \begin{align*} H\left( u\right) =-\mathbb{E}\left[ \frac{\partial _{\theta }\lambda _{t}^{\ast }\left( u\right) \left( \partial _{\theta }\lambda _{t}^{\ast }\left( u\right) \right) ^{\prime }}{\lambda _{t}^{\ast }(u)^{2}}\right] ,\;\ \ \Omega \left( u\right) =-\mathrm{Var}\left( \varepsilon _{t}^{2}\right) H\left( u\right). \end{align*} $$

If, in addition, $\mathbb {E}\left [ \varepsilon _{t}^{4+\delta }\right ] <\infty $ , for some $\delta>0$ , and $H\left ( x,\eta _{t},u\right ) $ satisfies Assumption 4.1(L3) and (L4) in Dahlhaus et al. (Reference Dahlhaus, Richter and Wu2019), then the results of Theorem 3 apply to the local constant estimators of the tv-ARCH-X model with $\partial _{u}H\left ( u\right ) =\partial H\left ( \theta |u\right ) /\left ( \partial u\right ) |_{\theta =\theta \left ( u\right ) }$ and

$$ \begin{align*} \partial _{\theta _{i}}H\left( u\right) =2\mathbb{E}\left[ \frac{\partial _{\theta _{i}}\lambda _{t}^{\ast }\left( u\right) \partial _{\theta }\lambda _{t}^{\ast }\left( u\right) \left( \partial _{\theta }\lambda _{t}^{\ast }\left( u\right) \right) ^{\prime }}{\lambda _{t}^{\ast }(u)^{3}}\right]. \end{align*} $$

Our conditions on the model are more or less identical to Bardet et al. (Reference Bardet, Doukhan and Wintenberger2022) but allows for exogenous regressors to be included. Our conditions are substantially weaker compared to Dahlhaus and Subba Rao (Reference Dahlhaus and Subba Rao2006) and Inoue, Jin, and Pelletier (Reference Inoue, Jin and Pelletier2019) who impose much stronger moment conditions.

Finally, we apply our results to the local MLE of the tv-PARX model in Example 3. Under (24) below, the tv-PARX process is locally stationary with stationary solution $Y_{t}^{\ast }(u)| \mathcal {F}_{t-1}^{\ast }(u) \sim \mathrm {Poisson}\left ( \lambda _{t}^{\ast }\left ( u\right ) \right ) $ , where $\lambda _{t}^{\ast }\left ( u\right ) =\theta \left ( u\right ) ^{\prime }\tilde {X}_{t}^{\ast }\left ( u\right ) $ and $\tilde {X}_{t}^{\ast }(u)=\left ( 1,Y_{t-1}^{\ast }(u),\ldots ,Y_{t-q}^{\ast }(u),X_{t-1}^{\ast }\left ( u\right ) ^{\prime }\right ) ^{\prime }$ .

Corollary 6. Suppose that Assumption 12 holds, $\Theta \subseteq \mathbb {R}_{+}^{1+q+d_{X}}$ is compact with $\omega \geq \underline {\omega }>0$ for all $\theta \in \Theta $ , $X_{n,t-1}\in \mathbb {R} _{+}^{d_{X}}$ , $\inf _{u\in \left [ 0,1\right ] }\omega \left ( u\right )>0,$ and

(24) $$ \begin{align} \sup_{u\in \left[ 0,1\right] }\sum_{i=1}^{q}\alpha _{i}\left( u\right) <1. \end{align} $$

Then, the results of Theorems 2 apply to the local linear estimators of the tv-PARX model (6) with

$$ \begin{align*} \Omega \left( u\right) =\mathbb{E}\left[ \frac{\left( \partial _{\theta }\lambda _{t}^{\ast }\left( u\right) \right) \left( \partial _{\theta }\lambda _{t}^{\ast }\left( u\right) \right) ^{\prime }}{\lambda _{t}^{\ast }\left( u\right) }\right] =-H\left( u\right) ,\;\ \ \partial _{\theta }\lambda _{t}^{\ast }\left( u\right) =\tilde{X}_{t}^{\ast }\left( u\right). \end{align*} $$

Suppose furthermore that $X_{t}^{\ast }\left ( u\right ) $ satisfies the following: Its Markov transition kernel is continuous w.r.t. the Lebesgue measure with density $p_{X}\left ( x|x_{0};u\right )>0$ which is differentiable w.r.t. u for all $\left ( x,x_{0}\right ) $ ; $\left \vert \partial _{u}p_{X}\left ( x|x_{0};u\right ) \right \vert /p_{X}\left ( x|x_{0};u\right ) \leq C\left ( 1+\left \Vert x\right \Vert +\left \Vert x_{0}\right \Vert \right ) $ ; $p_{X}\left ( x|x_{0};u\right ) \leq \bar {p} _{X}^{\left ( 0\right ) }\left ( x|x_{0};u\right ) $ and $\left \vert \partial _{u}p_{X}\left ( x|x_{0};u\right ) \right \vert \leq \left \vert \bar {p} _{X}^{\left ( 1\right ) }\left ( x|x_{0}\right ) \right \vert $ , where $\int \left \Vert x\right \Vert ^{2}\bar {p}_{X}^{\left ( k\right ) }\left ( x|x_{0};u\right ) dx<\infty $ , $k=0,1$ . Moreover, $\mathbb {E}\left [ 1+X_{t}^{\ast }\left ( u\right ) ^{2}|X_{t-1}^{\ast }\left ( u\right ) =x\right ] \leq \rho \left ( u\right ) \left ( 1+\left \Vert x\right \Vert ^{2}\right ) +b\left ( u\right ) $ for some $\rho \left ( u\right ) <1$ and $b\left ( u\right ) <\infty $ . Then the local constant estimator satisfies Theorem 3 with

$$ \begin{align*} \partial _{\theta _{i}}H\left( u\right) =2\mathbb{E}\left[ \frac{\partial _{\theta _{i}}\lambda _{t}^{\ast }\left( u\right) \partial _{\theta }\lambda _{t}^{\ast }\left( u\right) \left( \partial _{\theta }\lambda _{t}^{\ast }\left( u\right) \right) ^{\prime }}{\lambda _{t}^{\ast }(u)^{3}}\right]. \end{align*} $$

5 SIMULATION STUDY

In this section, we examine the finite-sample performances of the local constant and local linear estimators. All reported results are based on 1,000 simulated data sets. The overall performance of the estimators is evaluated using the mean absolute deviation error (MADE), $MADE_{i}:=\frac {1 }{n}\sum _{t=1}^{n}\mathbb {E}\left [ |\hat {\theta }_{i}\left ( t/n\right ) -\theta _{i}\left ( t/n\right ) |\right ] $ , as well as their integrated bias, variance, and mean squared error. All results are based on the Epanechnikov kernel and with the bandwidth chosen using the cross-validation method proposed in Richter and Dahlhaus (Reference Richter and Dahlhaus2019).

5.1 Time-Varying ARCH

We first consider the time-varying ARCH(1) in eq. (5), where $ \varepsilon \sim i.i.d.N\left ( 0,1\right ) $ , $\omega \left ( u\right ) =0.7-0.5\sin \left ( 4\pi u\right ) $ , and $\alpha \left ( u\right ) =0.45+0.4\sin \left ( 4\pi u\right ) $ . We estimate $\omega \left ( u\right ) $ and $\alpha \left ( u\right ) $ using both Gaussian log-likelihood and the WLS method of Fryzlewicz, Sapatinas, and Subba Rao (Reference Fryzlewicz, Sapatinas and Subba Rao2008). Table 1 reports the performance of the estimators. For all sample sizes, the local MLEs perform better than the local WLS estimators. For sample sizes of $n=250$ and $n=500$ , the local constant MLE performs as well as the local linear MLE in terms of the global measures, but the latter performs best in terms of IMSE and MADE for $n=1,000$ . Thus, the overall superiority of the local linear estimator indicated by the theory appears to be a large-sample property.

Table 1

Performance of the local constant (LC) and local linear (LL) estimators for tvARCH model: Integrated squared bias (IBias2), integrated variance (IVar), integrated mean squared errors (IMSE), and MADE.

Table 1 Long description

The table is structured with two main parameter columns: omega of u and alpha of u. Under each parameter, there are two estimation methods: W L S and M L. Each method is further divided into L C (Local Constant) and L L (Local Linear) estimators.

Data is organized by sample size n:

* For n = 250:

- omega of u: I Bias 2 ranges from 0.0202 to 0.0222; I Var from 0.0549 to 0.0787; I M S E from 0.0752 to 0.1009; M A D E from 0.2136 to 0.2386.

- alpha of u: I Bias 2 ranges from 0.0178 to 0.0279; I Var from 0.0720 to 0.0848; I M S E from 0.0933 to 0.1126; M A D E from 0.2379 to 0.2643.

* For n = 500:

- omega of u: I Bias 2 ranges from 0.0091 to 0.0118; I Var from 0.0313 to 0.0390; I M S E from 0.0405 to 0.0508; M A D E from 0.1555 to 0.1743.

- alpha of u: I Bias 2 ranges from 0.0076 to 0.0140; I Var from 0.0442 to 0.0484; I M S E from 0.0537 to 0.0624; M A D E from 0.1794 to 0.1977.

* For n = 1,000:

- omega of u: I Bias 2 ranges from 0.0043 to 0.0067; I Var from 0.0182 to 0.0225; I M S E from 0.0226 to 0.0285; M A D E from 0.1148 to 0.1301.

- alpha of u: I Bias 2 ranges from 0.0038 to 0.0072; I Var from 0.0257 to 0.0297; I M S E from 0.0306 to 0.0369; M A D E from 0.1353 to 0.1515.

An additional metric, I Bias 2 B D, consistently shows lower values for L L compared to L C across all sample sizes and parameters.

To compare the performance of the estimators near the end of the sample, we also evaluate the bias of the estimators for the first and last 2.5% of time periods corresponding to $u\in \left [ 0,0.025\right ] \cup \left [ 0.975,1 \right ] $ . This is reported as IBias2BD in Table 1. As predicted by the theory, we find that, relative to the local constant versions, the local linear WLS and ML estimators enjoy significantly smaller biases near the boundaries of $\left [ 0,1\right ] $ for all sample sizes.

5.2 Time-Varying Poisson Autoregression with Exogenous Covariates (PARX)

We here report simulation results for the local linear MLE of the following PARX(1) model with an additional exogenous regressor $X_{n,t}$ :

$$ \begin{align*} \lambda_{n,t}=\omega\left(t/n\right)+\alpha\left(t/n\right)Y_{n,t-1}+\gamma \left(t/n\right)X_{n,t-1}^{2}, \end{align*} $$

where $\omega \left (u\right )=0.6-0.3u+0.3\sin \left (2\pi u\right )$ , $ \alpha \left (u\right )=0.3+0.3u-0.3\sin \left (2\pi u\right )$ , and $ \gamma \left (u\right )=1-0.5\cos \left (\pi u\right )$ . The dynamics of the exogenous regressor was chosen as

$$ \begin{align*} X_{n,t}=\sqrt{\sigma\left(t/n\right)+\beta\left(t/n\right)X_{n,t-1}^{2}} \varepsilon_{t},\;\varepsilon_{t}\sim\text{i.i.d.}N\left(0,1\right). \end{align*} $$

where either

• • DGP1: $\sigma \left (u\right )=0.8$ and $\beta \left (u\right )=0.4$ so that $X_{n,t}=X_{t}$ is strictly stationary.

• • DGP2: $\sigma \left (u\right )=0.8+0.4\cos \left (2\pi u\right )$ and $ \beta \left (u\right )=0.4-0.2\sin \left (2\pi u\right )$ so $X_{n,t}$ is locally stationary.

Table 2 reports the overall performance of the estimators in terms of integrated squared bias, variance, MSE, and MADE. The table shows that, in all sample sizes, the local linear estimator behaves well for both DGP1 and DGP2. All bias, variance, and MADE decrease as the sample size increases. Finally, similar to the case of the tvARCH model, the local linear estimator performs well near the boundaries; we leave out these results since they are similar to the ones reported for the tv-ARCH model above.

Table 2

Performance of the local linear estimators for tvPARX models: Integrated squared bias (IBias2), integrated variance (IVar), integrated mean squared errors (IMSE), and median of MADE.

Table 2 Long description

The table compares performance metrics for parameters omega of u, alpha of u, and gamma of u across two data generating processes, D G P 1 and D G P 2. The metrics include I Bias 2, I Var, I M S E, and M A D E.

* For n = 250:

- D G P 1: omega of u has an I M S E of 0.1880; alpha of u is 0.0115; gamma of u is 0.0426.

- D G P 2: omega of u has an I M S E of 0.1748; alpha of u is 0.0111; gamma of u is 0.0386.

* For n = 500:

- D G P 1: omega of u has an I M S E of 0.0639; alpha of u is 0.0055; gamma of u is 0.0218.

- D G P 2: omega of u has an I M S E of 0.0784; alpha of u is 0.0057; gamma of u is 0.0209.

* For n = 1,000:

- D G P 1: omega of u has an I M S E of 0.0310; alpha of u is 0.0030; gamma of u is 0.0116.

- D G P 2: omega of u has an I M S E of 0.0330; alpha of u is 0.0030; gamma of u is 0.0112.

Across all parameters and both D G P s, I Bias 2, I Var, I M S E, and M A D E consistently decrease as the sample size n increases from 250 to 1,000.

6 EMPIRICAL APPLICATION

We here revisit the empirical analysis of U.S. corporate defaults carried out in Agosto et al. (Reference Agosto, Cavaliere, Kristensen and Rahbek2016) with the aim of examining whether there is evidence of structural instability in the time series. The data set consists of monthly number of bankruptcies among Moody’s rated industrial firms in the United States for the period 1982–2011 ( $n=360$ observations), collected from Moody’s Credit Risk Calculator (CRC). Figure 1 shows the time series of default counts together with its sample autocorrelation function, which reveals high temporal dependence in default counts and existence of default clusters over time.

Figure 1

Left: Number of defaults per month among Moody’s rated U.S. industrial firms in the period 1982–2011; Right: autocorrelation function of defaults.

Figure 1 Long description

The left panel is a line graph titled Actual Defaults. The horizontal x-axis represents Year from 1982 to 2011. The vertical y-axis ranges from 0 to 30. The blue line shows fluctuating default counts with significant peaks around 1990, 2001, and a major spike reaching nearly 30 around 2009. Four vertical gray shaded regions highlight specific periods of high default activity.

The right panel is a stem plot titled Sample Autocorrelation Function. The horizontal x-axis represents Lag from 0 to 20. The vertical y-axis ranges from -0.2 to 1. Red dots on vertical stems show the correlation values. At lag 0, the value is 1. The correlation decreases steadily in a non-linear decay as the lag increases, reaching approximately 0.1 at lag 20. Two horizontal blue lines indicate the significance thresholds around the 0.1 and -0.1 marks on the y-axis.

With $Y_{n,t}\in \left \{ 0,1,2,\ldots \right \} $ , $t\geq 1$ denoting the number of defaults in a given month, we use a log-PARX model to gain better understanding of the dynamics of $Y_{n,t}$ as a function of its own past, $ Y_{n,t-m}$ , $m\geq 1$ , but also in terms of additional covariates $ X_{n,t}\in \mathbb {R}^{d_{x}}$ , which include relevant macroeconomic and financial factors as considered in Agosto et al. (Reference Agosto, Cavaliere, Kristensen and Rahbek2016). We model $Y_{n,t}$ as a conditional Poisson distribution $Y_{n,t}|\mathcal {F}_{n,t-1}\sim \mathrm { Poisson}\left ( \lambda _{n,t}\left ( \theta \left ( t/n\right ) \right ) \right ) $ , $t=1,2,\ldots ,n$ , where the intensity $\lambda _{n,t}\left ( \theta \left ( t/n\right ) \right ) $ depends on past counts, covariates $X_{n,t-1}$ . and a vector of time-varying parameters $\theta \left ( t/n\right ) $ . Our favored specification of $\lambda _{n,t}\left ( \theta \right ) $ is the log-PARX model of Fokianos and Tjøstheim (Reference Fokianos and Tjøstheim2011), which is here augmented by the chosen set of exogenous variables, $X_{n,t-1}$ ,

(25) $$ \begin{align} \log \lambda _{n,t}\left( \theta \right) =\omega +\sum_{i=1}^{p}\alpha _{i}\log \left( Y_{n,t-i}+1\right) +\gamma ^{\prime }X_{n,t-1}, \end{align} $$

so that $\theta =\left ( \omega ,\alpha _{1},\ldots ,\alpha _{p},\gamma ^{\prime }\right ) ^{\prime }$ .

We here deviate from Agosto et al. (Reference Agosto, Cavaliere, Kristensen and Rahbek2016) that specifies $\lambda _{n,t}$ to be a linear function of past counts and factors. The reasons for us favoring the above log-specification over the linear one are three-fold: First, (25) do not impose positivity constraints on the parameters which facilitates the numerical computation of the estimators; second, it allows us to include any predictors we wish without the need of first transforming them to ensure that each component of the resulting $X_{n,t-1}$ is positive; third, when we estimate both models using the U.S. default data, we found that the log-specification delivers a better fit.

Similar to Agosto et al. (Reference Agosto, Cavaliere, Kristensen and Rahbek2016), we use $\sum _{i=1}^{p}\alpha _{i}\left ( t/n\right ) $ as a measure of contagion in the financial markets: If $ \sum _{i=1}^{p}\alpha _{i}\left ( t/n\right ) $ is large, then firms defaulting today will lead to a large increase in the risk of other firms defaulting next period everything else equal.

As exogenous covariates, we consider the same financial, credit market, and macroeconomic variables as in Agosto et al. (Reference Agosto, Cavaliere, Kristensen and Rahbek2016): Realized volatility ( $RV$ ) computed using daily squared return on the S&P 500 index, the Leading Index released by the Federal Reserve ( $LI$ ), year-to-year change in Industrial Production Index ( $IP$ ), one-year return on the S&P 500 index ( $SPX$ ), the three-month Treasury bill rate ( $TB3$ ), and BAA Moody’s rated to 10-year Treasury spread ( $SP$ ).Footnote ² Agosto et al. (Reference Agosto, Cavaliere, Kristensen and Rahbek2016) decompose $LI$ and $IP$ into their positive and negative parts to deal with above-mentioned issue of $X_{n,t-1}$ having to be positive in the linear specification. In contrast, no such transformations are needed for our log-specification. Moreover, as well as $RV$ , we also consider the logarithm of $RV$ , $\log \left ( RV\right ) $ , to evaluate if the latter is a better predictor; again, this would not be possible in the linear specification of Agosto et al. (Reference Agosto, Cavaliere, Kristensen and Rahbek2016).

The lag length p is chosen using BIC where, for a given model, the log-likelihood is evaluated at the estimated time-varying parameters.Footnote ³ According to this version of BIC, the preferred specification is $p=2$ . Importantly, by allowing for the parameters to be time-varying, a much more parsimonious model is selected by BIC: If we do model selection where for each model we restrict the estimated parameters to be constant over time, the preferred model is $p=6$ . Moreover, the persistence, or contagion, of the estimated time-invariant version, as measured by $\sum _{i=1}^{p}\hat {\alpha }_{i}$ , is substantially higher than for the time-varying version, as measured by $\sum _{i=1}^{p}\hat {\alpha } _{i}\left ( u\right ) $ , $u\in \left [ 0,1\right ] $ . This is consistent with the findings reported in Hillebrand (Reference Hillebrand2005) for GARCH models: Neglecting structural changes in parameters causes the estimates of these to be suffering from a strong upward bias which in turn leads BIC to selecting a bigger model.

As benchmark, we first estimate the time-invariant version of (25) with $p=2$ . Table 3 shows the estimation results for four different specifications of the time-invariant LLPARX(2) model: Column (1) contains the results when only $\left ( RV,LI\right ) $ are included; column (2) when only $\left ( \log RV,LI\right ) $ are included; column (3) when all covariates are included except for $\log \left ( RV\right ) $ ; and column (4) when all covariates except $RV$ are included. As in Agosto et al. (Reference Agosto, Cavaliere, Kristensen and Rahbek2016), once we control for the information contained in $RV$ and $LI$ , none of the other four covariates are found to be relevant in predicting future defaults. We also observe that the two specifications using $\log \left ( RV\right ) $ appear to perform better than the ones using $RV$ . Based on these results, our favored specification of the time-varying version is to use $LI$ and $\log \left ( RV\right ) $ as exogenous variables:

(26) $$ \begin{align} \begin{aligned} \log \lambda _{n,t}=\omega \left( t/n\right) & +\alpha _{1}\left( t/n\right) \log \left( Y_{n,t-1}+1\right) +\alpha _{2}\left( t/n\right) \log \left( Y_{n,t-2}+1\right) \\ & +\beta _{RV}\left( t/n\right) \log \left( RV_{n,t-1}\right) +\beta _{LI}\left( t/n\right) LI_{n,t-1}. \end{aligned} \end{align} $$

Table 3

Estimation results of different LLPARX models.

Table 3 Long description

The table consists of 5 columns (Parameter, Model 1, Model 2, Model 3, Model 4) and 10 parameter rows. Standard errors are provided in parentheses below each estimate.

* omega: Model 1 is 0.2186 (0.1022); Model 2 is 1.0425 (0.2491); Model 3 is 0.1003 (0.2118); Model 4 is 1.3015 (0.4188).

* alpha sub 1: Model 1 is 0.3040 (0.0492); Model 2 is 0.2777 (0.0496); Model 3 is 0.3093 (0.0497); Model 4 is 0.2836 (0.0501).

* alpha sub 2: Model 1 is 0.5033 (0.0491); Model 2 is 0.4876 (0.0486); Model 3 is 0.5082 (0.0499); Model 4 is 0.4956 (0.0492).

* R V: Model 1 is 5.8496 (3.9738); Model 3 is 5.4864 (4.4365). Models 2 and 4 are blank.

* log (R V): Model 2 is 0.1230 (0.0373); Model 4 is 0.1484 (0.0437). Models 1 and 3 are blank.

* L I: Model 1 is minus 0.1767 (0.0310); Model 2 is minus 0.1482 (0.0308); Model 3 is minus 0.1864 (0.0490); Model 4 is minus 0.1860 (0.0487).

* I P: Model 3 is 0.0029 (0.0480); Model 4 is minus 0.0167 (0.0481).

* S P W: Model 3 is 0.2074 (0.2408); Model 4 is 0.2412 (0.2413).

* R C 3: Model 3 is 0.0065 (0.0134); Model 4 is 0.0007 (0.0136).

* S Q: Model 3 is 0.0300 (0.0585); Model 4 is minus 0.0426 (0.0615).

Note: Standard errors are in parentheses.

Figure 2 shows the time series of the local linear estimates of the time-varying parameters of (26) together with the time-invariant estimates reported in column (2) of Table 3. Pointwise confidence bands are computed based on the asymptotic distribution derived in Theorem 3. The Leading Index is pointwise significant for most of the sample period which highlights the link between macroeconomic activity and corporate defaults also found in Agosto et al. (Reference Agosto, Cavaliere, Kristensen and Rahbek2016). At the same time, this link exhibits substantial time variation, in particular at the end of the sample during the Great Recession. The link between realized volatility and defaults of industrial firms is less significant but also appears to be changing over time. Similar to Agosto et al. (Reference Agosto, Cavaliere, Kristensen and Rahbek2016), the realized volatility and the Leading Index are strong explanatory variables during the Great Recession (2007–2011). However, differently, both of them are still relevant in the late 1980s and early 1990s. Especially, the effect of $\log \left ( RV\right ) $ tends to be negative in this period which cannot be captured by the aforementioned linear version of the PARX model.

Figure 2

Local linear estimate of time-varying parameter in eq. (26): Shaded areas are the 95% confidence intervals.

Figure 2 Long description

The figure consists of five panels arranged in two rows. Each panel plots a specific parameter estimate against the Year on the x-axis.

* Top-Left Panel: Plots omega hat (u) with tv L L P A R X (2). The y-axis ranges from minus 2 to 6. A blue line fluctuates significantly, starting near 4, dipping below 0 around 1992, and rising to 5 by 2012. A light blue shaded area represents the 95% C I. A horizontal red line sits at approximately 1.0.

* Top-Middle Panel: Plots alpha hat sub 1 (u). The y-axis ranges from minus 2.5 to 0.5. The blue line starts at minus 1.8 in 1982, rises to near 0 by 1990, and fluctuates slightly below the 0 dashed line until 2012. The red line is constant at 0.3.

* Top-Right Panel: Plots alpha hat sub 2 (u). The y-axis ranges from minus 0.5 to 0.5. The blue line peaks at 0.5 around 1986 and then trends downward with fluctuations toward minus 0.3. The red line is constant at 0.5.

* Bottom-Left Panel: Plots gamma hat sub R V (u). The y-axis ranges from minus 0.4 to 0.8. The blue line shows a wave-like pattern, dipping to minus 0.2 in 1992 and rising to 0.3 by 2012. The red line is constant at 0.1.

* Bottom-Right Panel: Plots gamma hat sub L I (u). The y-axis ranges from minus 2 to 0. The blue line stays between minus 0.5 and 0 until 2007, where it peaks slightly above 0 before sharply dropping to minus 2.0 by 2012. The red line is constant at minus 0.2.

All panels feature vertical grey shaded bars indicating specific time intervals and a dashed black horizontal line at y equals 0.

Finally, judging from $\hat {\alpha }_{1}\left ( t/n\right ) +\hat {\alpha } _{2}\left ( t/n\right ) $ , there is very little contagion in the financial markets from the 1990s onward, which is somewhat consistent with the findings in Agosto et al. (Reference Agosto, Cavaliere, Kristensen and Rahbek2016). However, at the end of the sample, we actually find a negative effect of today’s log-defaults on tomorrow’s default risk. Also note that the time-varying estimates, $\hat {\alpha } _{1}\left ( t/n\right ) $ and $\hat {\alpha }_{2}\left ( t/n\right ) $ , remain below the corresponding time-invariant ones, $\hat {\alpha }_{1}$ and $\hat { \alpha }_{2}$ as marked by the horizontal red lines, throughout the sample period. Again, this seems to indicate that ignoring time variation in the parameters of PARX models lead to over estimation of the level of persistence/contagion.

To assess in-sample fit and whether the reported time variation in the parameters is statistically significant, we carry out an array of graphical and quantitative diagnostic tools for time series. First, we plot in the left panel of Figure 3 the actual default counts together with the predicted defaults $\hat {Y}_{n,t}:=\hat {\lambda } _{n,t}=\lambda _{n,t}\left ( \hat {\theta }\left ( t/n\right ) \right ) $ . As can be seen from this plot, the time-varying LLPARX model captures the default counts dynamics well. In the right panel of Figure 3, the sample autocorrelation function of the standardized Pearson residuals $ \hat {e}_{n,t}=\hat {\lambda }_{n,t}^{-1/2}\left ( Y_{n,t}-\hat {\lambda } _{n,t}\right ) $ is plotted. Under correct specification, $e_{n,t}$ should be white noise—the plotted sample autocorrelation function supports this.

Figure 3

Left: Actual number of defaults (blue) and estimated intensity (red); Right: Sample autocorrelation function of Pearson residuals.

Figure 3 Long description

The figure consists of two panels arranged horizontally.

Left Panel: A line graph showing the time series of defaults from 1982 to 2012.

* The X axis is labeled Year, ranging from 1982 to 2012.

* The Y axis represents the count, ranging from 0 to 30.

* A blue line represents Actual Defaults, showing a highly volatile path with major peaks around 1991, 2002, and a primary spike exceeding 25 in 2009.

* A red line represents Estimated Intensity, which follows the blue line closely but with a smoother, less volatile trajectory.

* Four vertical grey shaded regions highlight specific periods of high default activity around 1982, 1991, 2001 to 2002, and 2008 to 2009.

Right Panel: A stem plot showing the sample A C F of Pearson residuals.

* The X axis is labeled Lag, ranging from 0 to 20.

* The Y axis ranges from minus 0.2 to 1.0.

* A legend at the top right identifies the data as t v L L P A R X 2 model Residual Autocorrelation Function.

* At Lag 0, the autocorrelation is exactly 1.0.

* For Lags 1 through 20, the red dots (stems) fluctuate near the zero line, almost entirely contained within the blue horizontal significance bounds located at approximately plus or minus 0.1.

We also evaluate the adequacy of fit using the probability integral transform (PIT). We follow Davis and Liu (Reference Davis and Liu2016) and compute the PITs by

$$ \begin{align*} \hat{u}_{n,t}:=F_{n,t}\left( Y_{n,t}-1\right) +\nu _{t}\left[ F_{n,t}\left( Y_{n,t}\right) -F_{n,t}\left( Y_{n,t}-1\right) \right] , \end{align*} $$

where $\left \{ \nu _{t}\right \} $ is a sequence of i.i.d. random variables from a standard uniform distribution, and $F_{n,t}$ is the CDF of a Poisson( $ \hat {\lambda }_{n,t})$ distribution. Under correct model specification, $\hat { u}_{n,t}$ is a sequence of i.i.d. random variables from the standard uniform distribution. Figure 4 depicts the histogram of PIT which show that the tvLLPARX(2) model provides a better in-sample fit than the corresponding time-invariant model.

Figure 4

Left: Histograms of randomized PITs for log-linear PARX(2) and time-varying log-linear PARX(2) models fitted to the U.S. default data; Right: QQ-plots of the randomized PIT against standard uniform distribution for the corresponding models.

Figure 4 Long description

The figure consists of four panels arranged in a two-by-two grid.

Top-Left Panel: A histogram titled P I T of L L P A R X (2). The x-axis ranges from 0 to 1 in increments of 0.1, and the y-axis represents frequency from 0 to 50. The distribution is U-shaped, with the highest bars at the 0 to 0.1 and 0.9 to 1 intervals, and a trough in the center around 0.4 to 0.5.

Top-Right Panel: A Q Q plot titled Q Q-plot of P I T for L L P A R X (2). The x-axis and y-axis both range from 0 to 1. A blue dotted diagonal reference line represents a standard uniform distribution. An orange solid line representing the model data follows the diagonal but shows a slight S-shaped deviation, dipping below the line in the lower half and rising above it in the upper half.

Bottom-Left Panel: A histogram titled P I T of t v L L P A R X (2). The axes are identical to the top-left panel. The distribution is more uniform than the top-left, though it still shows a slight peak at the 0.9 to 1 interval and a minor dip between 0.4 and 0.7.

Bottom-Right Panel: A Q Q plot titled Q Q-plot of P I T for t v L L P A R X (2). The orange solid line follows the blue dotted diagonal reference line much more closely than the top-right plot, indicating a better fit to the standard uniform distribution.

Finally, in Table 4, we report the log-likelihood, AIC, and BIC values and the p-value from a Kolmogorov–Smirnov test of the PITs being uniformly distributed of each of four specifications in columns (1)–(4) of Table 3 together with the time-varying versions of (2) and (4), labeled tv (2) and tv(4), respectively. From these, we see that allowing for time-varying parameters increase the in-sample fit dramatically. While this is not a formal statistical test of time variation, it provides strong informal evidence of such. As mentioned earlier, we also see that specification (2) is favored over (1), (3), and (4) in the time-invariant case, and that (2) is favored over (4) in the time-varying case.

Table 4

In-sample fit of time-invariant and time-varying LLPARX models.

Table 4 Long description

The table consists of seven columns. The first column lists the metrics, while columns two through seven are labeled (1), (2), tv (2), (3), (4), and tv (4).

* Row 1, log L: (1) 696.3, (2) 700.6, tv (2) 777.7, (3) 696.8, (4) 701.8, tv (4) 778.6.

* Row 2, A I C: (1) minus 1,382.5, (2) minus 1,391.2, tv (2) minus 1,526.6, (3) minus 1,375.6, (4) minus 1,385.5, tv (4) minus 1,539.3.

* Row 3, B I C: (1) minus 1,363.1, (2) minus 1,371.7, tv (2) minus 1,507.2, (3) minus 1,340.6, (4) minus 1,350.5, tv (4) minus 1,504.3.

* Row 4, p-value of P I T: (1) 0.0726, (2) 0.1319, tv (2) 0.5432, (3) 0.0588, (4) 0.1544, tv (4) 0.6445.

We complete the analysis by conducting a final sensitivity analysis of (26). This is done by including the remaining exogenous covariates in addition to realized volatility and Leading Index in the time-varying version of the model:

(27) $$ \begin{align} \begin{aligned} \log \lambda _{n,t}=\omega \left( t/n\right) & +\alpha _{1}\left( t/n\right) \log \left( Y_{n,t-1}+1\right) +\alpha _{2}\left( t/n\right) \log \left( Y_{n,t-2}+1\right) \\ & +\gamma _{RV}\left( t/n\right) \log \left( RV_{n,t-1}\right) +\gamma _{LI}\left( t/n\right) LI_{n,t-1}+\gamma _{IP}\left( t/n\right) IP_{n,t-1} \\ & +\gamma _{SPX}\left( t/n\right) SPX_{n,t-1}+\gamma_{TB3}\left( t/n\right) TB3_{n,t-1}+\gamma_{SP}\left( t/n\right) SP_{n,t-1}. \end{aligned} \end{align} $$

The estimation results are provided in Figure 5. The estimates, except for the realized volatility, are consistent with our baseline findings. The Leading Index remains highly significant, which shows that macroeconomic factors are relevant in predicting future defaults. The link between short-term interest rates and defaults of industrial firms changes over time. During the late 1980s and the Great Recession (2007–2011), interest rates play a role in determining the interest expense of firms. In the 1990s, similar to the finding in Duffie et al. (Reference Duffie, Saita and Wang2007), the sign of the coefficient for the short-term rate is consistent with the fact that the U.S. Federal Reserve often increases the short-term rates to control business expansions. Controlling for $LI$ and $TB3$ , other covariates are estimated to be insignificant for most of the sample period except for the period of the Great Recession (2007–2011), in which financial, credit market, and macroeconomic variables are significant explanators of the default intensity. These are novel findings that the original analysis of Agosto et al. (Reference Agosto, Cavaliere, Kristensen and Rahbek2016) did not reveal.

Figure 5

Local linear estimate of time-varying parameter in eq. (27): Shaded areas are the 95% confidence intervals.

Figure 5 Long description

The figure consists of nine panels arranged in three rows and three columns. Each panel features a horizontal x-axis representing Year, ranging from approximately 1982 to 2010. The y-axis represents various parameter estimates. Each graph contains three main elements: a fluctuating blue line representing the time-varying local linear estimate (t v L L P A R X 2), a light blue shaded region representing the 95 percent confidence interval (C I), and a solid horizontal red line representing the constant estimate (L L P A R X 2). A dashed black line at zero serves as a baseline.

* Top-left panel: Parameter omega hat (u). The blue line fluctuates around the red line, which is set near 2.

* Top-center panel: Parameter alpha hat sub 1 (u). The blue line starts low at -1 and rises toward the red line at 0.3.

* Top-right panel: Parameter alpha hat sub 2 (u). The blue line fluctuates between 0 and 0.4, while the red line is at 0.5.

* Middle-left panel: Parameter gamma hat sub R V (u). The blue line shows a downward trend from 0.2 to -0.1, while the red line is constant at 0.15.

* Middle-center panel: Parameter gamma hat sub L I (u). The blue line shows a significant decline from 0 to -1.5 after 2007, while the red line is at -0.2.

* Middle-right panel: Parameter gamma hat sub S P X (u). The blue line is relatively stable near 0 until a sharp spike to 4 in 2010, while the red line is at 0.2.

* Bottom-left panel: Parameter gamma hat sub I P (u). The blue line oscillates around the red line and zero baseline.

* Bottom-center panel: Parameter gamma hat sub T B 3 (u). The blue line shows high volatility with a sharp dip in 1995 and a spike in 2010.

* Bottom-right panel: Parameter gamma hat sub S P (u). The blue line remains near 0 for most of the period before a vertical spike to 3.5 at the end of the timeline.

7 CONCLUSION AND FUTURE WORK

We have here developed a general theory for kernel regression-type estimators of time-varying parameters and used this to provide primitive conditions for the estimators in Markov models. Another interesting application would be infinite memory models of the form

$$ \begin{align*} Y_{n,t}=G\left( Y_{n,t-1},Y_{n,t-2},\dots,\varepsilon _{t};\theta \left( t/n\right) \right). \end{align*} $$

The local constant estimator of $\theta \left ( u\right ) $ was analyzed in Bardet et al. (Reference Bardet, Doukhan and Wintenberger2022) using techniques different from ours. Specifically, they provide primitive conditions under which $Y_{n,t}$ satisfies

$$ \begin{align*} \sup_{\left[ n\left( u-cb\right) \right] \leq t\leq \left[ n\left( u-cb\right) \right] +2cnb}E\left[ \left\Vert Y_{n,t}-Y_{t}^{\ast }\left( u\right) \right\Vert ^{p}\right] ^{1/p}\leq C\left( b^{\rho }+\lambda _{ \left[ n\left( u-cb\right) \right] +t}\right), \end{align*} $$

where $Y_{t}^{\ast }\left ( u\right ) $ is the stationary solution to $ Y_{t}^{\ast }\left ( u\right ) =G\left ( Y_{t-1}^{\ast }\left ( u\right ) ,Y_{t-2}^{\ast }\left ( u\right ) ,\dots ,\varepsilon _{t};]\right.\left. \theta \left ( u\right ) \right ) $ . But this result unfortunately does not suffice for our theory since this requires $Y_{n,t}$ to satisfy our version of local stationarity as given in Definition 1. At the same time, Bardet et al. (Reference Bardet, Doukhan and Wintenberger2022) do provide primitive conditions for the assumptions on the stationary version of Corollary 3 to hold. Thus, under the high-level assumption that $Y_{n,t}$ satisfies our version of local stationarity, we can combine the results of Bardet et al. (Reference Bardet, Doukhan and Wintenberger2022) with our Corollary 3 to obtain a theory for local polynomial estimators of infinite memory models. Under this high-level condition, we are also able to provide a precise characterization of the leading bias term of their estimator, something which was missing in the analysis of Bardet et al. (Reference Bardet, Doukhan and Wintenberger2022). We leave it to future research to establish primitive conditions under which infinite memory models satisfy our Definition 1.