Common large innovations across nonlinear time series
-
Philip Hans Franses
and
Richard Paap
Abstract
We propose a multivariate nonlinear econometric time series model, which can be used to examine if there is common nonlinearity across economic variables. The model is a multivariate censored latent effects autoregression. The key feature of this model is that nonlinearity appears as separate innovation-like variables. Common nonlinearity can then be easily defined as the presence of common innovations. We discuss representation, inference, estimation and diagnostics. We illustrate the model for US and Canadian unemployment and find that US innovation variables have an effect on Canadian unemployment, and not the other way around, and also that there is no common nonlinearity across the unemployment variables.
Appendix
The relevant part of the integrands in (15) can be written in the form
where ut=(u1t,u2t) and 
for g01, 
for g10 and 
for g11. Furthermore, 
with 
for g01, 
for g10, 
for g11. If we write D=(D1|D2) where Di denotes the ith column of D, we have that St=–(0|D2) for g01, St=–(D1|0) for g10 and St=–D for g11. To simplify the computation of the three integrals in the likelihood function we use the following general result
with 
Hence (30) equals
This result allows us to write the integrands in the three integrals in (15) as the product of a normal density function which depends on ut and a remainder that does not depend on ut. Hence, the integral can be expressed in terms of the distribution function of a (bivariate) normal distribution with mean bt and (co)variance matrix 
which is available in most packages.
- 1
It is easy to see that σu12 is not identified if either v1t or v2t is positive. Hence, if one wants to allow for correlation between u1t and u2t its identification will be based on observations where both v1t and v2t are zero in which case there is limited information about the values of u1t and u2t Unreported estimation results show that the covariance is empirically badly identified. Therefore we opt for σu12.
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Articles in the same Issue
- Masthead
- Masthead
- The Danish krone-euro exchange rate and Danmark Nationalbank intervention operations
- Common large innovations across nonlinear time series
- The forward rate premium puzzle: a case of misspecification?1)
- A smooth transition long-memory model
- Nonlinear causality tests and multivariate conditional heteroskedasticity: a simulation study
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Articles in the same Issue
- Masthead
- Masthead
- The Danish krone-euro exchange rate and Danmark Nationalbank intervention operations
- Common large innovations across nonlinear time series
- The forward rate premium puzzle: a case of misspecification?1)
- A smooth transition long-memory model
- Nonlinear causality tests and multivariate conditional heteroskedasticity: a simulation study
- Threshold linkages between volatility and trading volume: evidence from developed and emerging markets
- Inventory investment and the business cycle: the usual suspect
