This function allows the user to test linearity against a Vector Smooth Transition Autoregressive Model with a single transition variable.
VLSTARjoint(y, exo = NULL, st, st.choice = FALSE, alpha = 0.05)data.frame or matrix of dependent variables of dimension (Txn)
(optional) data.frame or matrix of exogenous variables of dimension (Txk)
single transition variable for all the equation of dimension (Tx1)
boolean identifying whether the transition variable should be selected from a matrix of R potential variables of dimension (TxR)
Confidence level
An object of class VLSTARjoint.
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.
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