VLSTARjoint: Joint linearity test
In starvars: Vector Logistic Smooth Transition Models Estimation and Prediction

Description Usage Arguments Details Value Author(s) References

This function allows the user to test linearity against a Vector Smooth Transition Autoregressive Model with a single transition variable.

1	VLSTARjoint(y, exo = NULL, st, st.choice = FALSE, alpha = 0.05)

`y`	`data.frame` or `matrix` of dependent variables of dimension `(Txn)`
`exo`	(optional) `data.frame` or `matrix` of exogenous variables of dimension `(Txk)`
`st`	single transition variable for all the equation of dimension `(Tx1)`
`st.choice`	boolean identifying whether the transition variable should be selected from a matrix of `R` potential variables of dimension `(TxR)`
`alpha`	Confidence level

Given a VLSTAR model with a unique transition variable, s_{1t} = s_{2t} = … = 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} : γ_{j} = 0, j = 1, …, \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 γ_{j} = 0.

The approximation of the logistic function with a first-order Taylor expansion is given by

G(s_t; γ_{j},c_{j}) = (1/2) + (1/4)γ_{j}(s_t-c_{j}) + r_{jt}

= a_{j}s_t + b_{j} + r_{jt}

where a_{j} = γ_{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}, …, a_{\widetilde{n}}s_t+b_{\widetilde{n}} + r_{\widetilde{n}t}\big\}

= As_t + B + R_t

where A = diag(a_{1}, …, a_{\widetilde{n}}), B = diag(b_{1},…, b_{\widetilde{n}}) e R_t = diag(r_{1t}, …, 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

= Θ_{0}z_t + Θ_{1}z_ts_t+\varepsilon_t^{*}

where Θ_{0} = B_{1} + B_{2}'B, Θ_{1} = B_{2}'A and \varepsilon_t^{*} = R_tB_{2} + \varepsilon_t. Under the null, Θ_{0} = B_{1} and Θ_{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{θ})}{\partial Θ_{1}} = ∑_{t=1}^{T}z_ts_t(y_t - \widetilde{B}_{1}z_t)'\widetilde{Ω}^{-1} = S(Y - Z\widetilde{B}_{1})\widetilde{Ω}^{-1},

where

S = z_{1}'s_{1}\\\vdots\\ z_t's_t

and where \widetilde{B}_{1} and \widetilde{Ω} 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{Ω}^{-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 χ^{2} with \widetilde{n}(p\cdot\widetilde{n} + k) degrees of freedom.

An object of class VLSTARjoint.

Andrea Bucci

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

starvars documentation built on Jan. 18, 2022, 1:08 a.m.