面板阈值模型

d
(6.64)
as Under certain assumptions and H 0 : 0 , Hansen shows that LR1
N , where is a random variable with distribution function
28
and identically distributed (iid) with mean zero and finite variance 2 . The iid assumption excludes lagged dependent variables from xit .The analysis is asymptotic with fixed T as
t 1
T
T
t 1
Y * X * e*
(6.55)
For any given , the slope coefficient can be estimated by ordinary least squares (OLS). That is,
ˆ X * X *
H 0 : 1 2
Under the null hypothesis of no threshold, the model is
yit i 1 xit eit
After the fixed-effect transformation is made, we have
* * * yit 1 xit eit
* yit x* e* it it
(6.54)
where
* yit yit yi ,
* xit xit xi ,
* eit eit ei
and
yi T 1 yit ,
t 1
T
xi T 1 xit , ei T 1 eit . Using the matrix notation, (6.54) is equivalent to
(6.63)
The asymptotic distribution of F1 is non-standard. Hansen (1996) shows that a bootstrap procedure attains the first-order asymptotic distribution, so p-values constructed from the bootstrap are asymptotically valid. (2) Asymptotic distribution of threshold estimate When there is a threshold effect
1 2

so that (6.52) equals
yit i xit eit
(6.53)
The observations are divided into two ‘regimes’ depending on whether the threshold variable qit is smaller or larger than the threshold . The regimes are distinguished by differing regression slopes, 1 and 2 . For the identification of 1 and 2 , it is required that the elements of xit are not time invariant. We also assume that the threshold variable qit is not time invariant. The error eit iatistic for tests on . To test the hypothesis H 0 : 0 , the likelihood ratio test is to reject for large values of LR1 0 where
ˆ2 LR1 S1 S1 ˆ
1 xit eit , qit . yit i x e , q 2 it it it i
Another compact representation of (6.52) is to set xit
xit I qit and x I q it it
N .
6.3.2 Model Estiamtion One traditional method to eliminate the individual effect
i is to remove
individual-specific means. Note that taking averages of (6.52) over the time index t and taking the difference of (6.52) and the averages produces
(6.61)
(6.62)
* * * By OLS , we have 1 、 eit and the sum of squared errors S 0 e e . The likelihood
ratio test of H0 is based on
ˆ ˆ2 F1 S0 S1
1 2 Chan (1993) and Hansen (2000) have
ˆ is consistent for 0 ( the true value of ) and that the asymptotic shown that
distribution is highly non-standard. Hansen (1999) argues that the best way to form confidence intervals for is to form the ‘no-rejection region' using the likelihood ratio
P x 1 exp x 2
2
(6.65)
Since the asymptotic distribution is pivotal, i.e., this limiting distribution does not depend on nuisance parameters, it may be used to form valid asymptotic confidence intervals. Furthermore, the distribution function (6.65) has the inverse
c 2log 1 1


(6.66)
from which it is easy to calculate critical values. For example, the 5% critical value is 7.35. A test of H 0 : 0 rejects at the asymptotic level
6.3 Threshold Effects in Non-dynamic Panels
Are regression functions identical across all observations in a sample, or do they fall into discrete classes? This question may be addressed using threshold regression techniques. Threshold regression models specify that individual observations can be divided into classes based on the value of an observed variable. Hansen (1999) introduces econometric techniques appropriate for threshold regression with panel data. 6.3.1 Model Setup The observed data are from a balance panel
X * Y * Chan (1993) and Hansen (2000) recommend estimation of by least squares. This
1
Y * I X * X * X *


(6.59)
ˆ ˆ ˆ is obtained, the slope coefficient estimate is ˆ . The residual Once
ˆ e ˆ vector is e
*
*
ˆ and residual variance
ˆ2 1 1 ˆ*e ˆ* e S1 ˆ N (T 1) N (T 1)
(6.60)
6.3.3 Inference (1) Testing for a threshold It is important to determine whether the threshold effect is statistically significant. The hypothesis of no threshold effect in (6.52) can be represented by the linear constraint