| VLSTARjoint {starvars} | R Documentation |
Joint linearity test
Description
This function allows the user to test linearity against a Vector Smooth Transition Autoregressive Model with a single transition variable.
Usage
VLSTARjoint(y, exo = NULL, st, st.choice = FALSE, alpha = 0.05)
Arguments
y |
|
exo |
(optional) |
st |
single transition variable for all the equation of dimension |
st.choice |
boolean identifying whether the transition variable should be selected from a matrix of |
alpha |
Confidence level |
Details
Given a VLSTAR model with a unique transition variable, s_{1t} = s_{2t} = \dots = s_{\widetilde{n}t} = s_t, a generalization of the
linearity test presented in Luukkonen, Saikkonen and Terasvirta (1988) may be implemented.
Assuming a 2-state VLSTAR model, such that
y_t = B_{1}z_t + G_tB_{2}z_t + \varepsilon_t.
Where the null H_{0} : \gamma_{j} = 0, j = 1, \dots, \widetilde{n}, is such that G_t \equiv (1/2)/\widetilde{n} and the
previous Equation is linear. When the null cannot be rejected, an identification problem of the parameter c_{j} in the transition function
emerges, that can be solved through a first-order Taylor expansion around \gamma_{j} = 0.
The approximation of the logistic function with a first-order Taylor expansion is given by
G(s_t; \gamma_{j},c_{j}) = (1/2) + (1/4)\gamma_{j}(s_t-c_{j}) + r_{jt}
= a_{j}s_t + b_{j} + r_{jt}
where a_{j} = \gamma_{j}/4, b_{j} = 1/2 - a_{j}c_{j} and r_{j} is the error of the approximation. If G_t is specified as follows
G_t = diag\big\{a_{1}s_t + b_{1} + r_{1t}, \dots, a_{\widetilde{n}}s_t+b_{\widetilde{n}} + r_{\widetilde{n}t}\big\}
= As_t + B + R_t
where A = diag(a_{1}, \dots, a_{\widetilde{n}}), B = diag(b_{1},\dots, b_{\widetilde{n}}) e R_t = diag(r_{1t}, \dots,
r_{\widetilde{n}t}), y_t can be written as
y_t = B_{1}z_t + (As_t + B + R_t)B_{2}z_t+\varepsilon_t
= (B_{1} + BB_{2})z_t+AB_{2}z_ts_t + R_tB_{2}z_t + \varepsilon_t
= \Theta_{0}z_t + \Theta_{1}z_ts_t+\varepsilon_t^{*}
where \Theta_{0} = B_{1} + B_{2}'B, \Theta_{1} = B_{2}'A and \varepsilon_t^{*} = R_tB_{2} + \varepsilon_t. Under the null,
\Theta_{0} = B_{1} and \Theta_{1} = 0, while the previous model is linear, with \varepsilon_t^{*} = \varepsilon_t. It
follows that the Lagrange multiplier test, under the null, is derived from the score
\frac{\partial \log L(\widetilde{\theta})}{\partial \Theta_{1}} = \sum_{t=1}^{T}z_ts_t(y_t - \widetilde{B}_{1}z_t)'\widetilde{\Omega}^{-1} = S(Y - Z\widetilde{B}_{1})\widetilde{\Omega}^{-1},
where
S = z_{1}'s_{1}\\\vdots\\ z_t's_t
and where \widetilde{B}_{1} and \widetilde{\Omega} are estimated from the model in H_{0}. If P_{Z} = Z(Z'Z)^{-1}Z' is the
projection matrix of Z, the LM test is specified as follows
LM = tr\big\{\widetilde{\Omega}^{-1}(Y - Z\widetilde{B}_{1})'S\big[S'(I_t - P_{Z})S\big]^{-1}S'(Y-Z\widetilde{B}_{1})\big\}.
Under the null, the test statistics is distributed as a \chi^{2} with \widetilde{n}(p\cdot\widetilde{n} + k) degrees of freedom.
Value
An object of class VLSTARjoint.
Author(s)
Andrea Bucci
References
Luukkonen R., Saikkonen P. and Terasvirta T. (1988), Testing Linearity Against Smooth Transition Autoregressive Models. Biometrika, 75: 491-499
Terasvirta T. and Yang Y. (2015), Linearity and Misspecification Tests for Vector Smooth Transition Regression Models. CREATES Research Paper 2014-4