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From Reviews: 0MR4215856 62G08 91B84
Wang, Qiying [Wang, Qi Ying] (5-SYD-SM) ;Phillips, Peter C. B. (1-YALE-NDM) ; Kasparis, Ioannis (CY-CYP-NDM)
Latent variable nonparametric cointegrating regression. (English summary)
Econometric Theory 37 (2021), no. 1, 138–168.

This article investigates statistical properties of a nonparametric cointegrating regression model with a latent regressor. Let (yt , zt ) be two nonstationary variables that arelinked according to the nonparametric regression modelyt = g(zt ) + ηt ,t = 1, . . . , n,where g(·) is an unknown function and ηt is a zero-mean stationary disturbance that
may be correlated with zt . Throughout it is assumed that the true regressor zt isunobservable, whereas its proxy xt is available. Such circumstances arise when zt ismeasured with error or given in filtered forms as in the mixed-data-sampling (MIDAS)
regression or Hodrick-Prescott (HP) filtering.
In this scenario, it is a common practice to use xt in place of zt to estimate g(x)nonparametrically asPn
yt K{(xt − x)h}bg (x) = Pt=1,nt=1 K{(xt − x)/h}
where K(·) is a nonnegative kernel function and h is a bandwidth. To explore the limit
theory of bg (x), observe thatPnPn
t=1 g(zt )K{(xt − x)/h}t=1 ηt K{(xt − x)/h}
P+ P.
(1)
bg (x) =nnK{(x−x)/h}t
t=1
t=1 K{(xt − x)/h}
While the first term can be handled
Pnin a manner analogous to correctly specified cases,
the second term involves Sn := t=1 g(zt )K{(xt − x)/h}, the limit theory of which
depends on the linkage between two nonstationary variables, (xt , zt ), and n, and is new
in the literature.
To obtain a general result on the limit behavior of Sn , let at be asymptoticallycointegrated with xt so thatat = γnt xt + wnt + ut ,where γnt is a sample-size-dependent coefficient with a limit γ0 (e.g., an estimatedcoefficient), wnt is an asymptotically negligible misspecification error,t is ap andPunstationary disturbance. Also let ξt be the innovation for xt and dn := Var( t=1 ξt2 ).
Then, it is demonstrated that for a class of continuous functions f (·), a given x and thebandwidth satisfying h + dn /(nh) → 0 as n → ∞,ndn X
xt − x d(2)f (at )K
→ E{f (γ0 x + u1 )}LZ (1, 0),
nh t=1hwhere LZ (t, x) is the continuous local time for a fractional Ornstein-Uhlenbeck process
Zt .
The limit behavior of bg (x) can be obtained by applying (2) to the second term of (1).
Suppose that (xt , zt ) has the linkage
xt = αnt zt + wnt + ut ,−1where max1≤k≤n |αnk
− 1| → 0 as n → ∞, and xt (and thus zt ) is correlated with ηt .
Then, the first term of (1) is shown to be Op {(nh2 )1/4 }, whereas reading f as g in (2)yields()nn
xt − xdn X
xt − xdn X
d→ {LZ (1, 0), E{g(x − u1 )}LZ (1, 0)}.
K,g(zt )K
nh t=1hnh t=1hTherefore, applying the continuous mapping theorem to the second term of (1), wefinally havepbg (x) = Op {(nh2 )1/4 } + E{g(x − u1 )} + op (1) → E{g(x − u1 )} =: g1 (x).
The function g1 (x) is a weighted average of g locally around x with weights determinedby the stationary distribution of the measurement error u. Also observe that bg (x)is usually inconsistent because g(x) 6= g1 (x) in general. However, when g is linear,g(x) = g1 (x) holds and thus bg (x) becomes consistent. This corresponds to the case inwhich the true regressor zt is measured with a stationary error in a linear cointegrating
regression.
REVISED (December, 2021)
Current version of review. Go to earlier version.
Masayuki Hirukawac Copyright American Mathematical Society 2021