0 50 100 150 200 250 70 80 90 100 110 Exchange r ate 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 Lag A CF 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 Lag P ar tial A CF
Figure 2.17: Graph, ACF and PACF of the effective exchange rate for euro area.
Normality ARCH effects Serial correlation
JB A(1) A(4) A(8) LM(8) LM(12) LM(24)
p-value 0.000 0.258 0.144 0.202 0.874 0.973 0.868
JB is the Jarque-Bera test, A(i)is the ithorder ARCH test, LM(i)denotes the ithorder Breusch-Godfrey LM test.
Table 2.10: Diagnostic tests for residuals. We assume that the conditional mean of the time series is described by a continuous time AR(1) model.
0 5 10 15 20 0.0 0.4 0.8 Lag Residuals 0 5 10 15 20 0.0 0.4 0.8 Lag Squared Residuals
Figure 2.18: ACF of residuals and squared residuals. We assume that the conditional mean of the time series is described by a continuous time AR(1) model.
Chapter 3
Cointegrated continuous time
models with mixed sample data
3.1
Introduction
Continuous time models have been commonly used to describe the dynamics of economic behaviour. Although information is collected at certain frequencies such as quarters, months or years, the movements in the variables do not necessarily coincide with the observational interval, this being the case of macroeconomic information, which is collected at low frequencies (usually quarterly or annually). Continuous time modelling may, therefore, provide a more realistic approximation to the actual dynamics of the economy, which involve a large number of economic agents making decisions at different points of time.
The advantages of this type of models over those formulated in discrete time have been widely discussed in the literature (see for example, Bergstrom, 1996; Bergstrom and Nowman, 2007). Among them is the separate treatment of stock and flow variables. It is explicitly recognised that stocks are observed at specific points of time, while flows are measured as the accumulation of the underlying rate over a time interval. Because of the distinction in the treatment of these two type of variables, these models do not suffer from time aggregation bias whereas it may be a serious problem in their discrete time counterpart.
Continuous time models also have the advantage of allowing for a mixture of both stock and flow variables. Mixed sampling often arises in econometric modelling. A typical money demand model, for example, comprises variables that are measured instantaneously such as prices and exchange rates, and variables that are observed as integrals such as income. Another example is the Fisher effect, this hypothesis being
tested by using interest rates (stocks), and expected or actual inflation rate (a flow). Different approaches have been considered to estimate parameters of continuous time systems with mixed sample data. Among them are Kalman filtering techniques based on state space representations (see Harvey and Stock, 1985, 1988; Zadrozny, 1988), and methods based on spectral representations (see Phillips, 1991a; Robinson, 1993). Another important contribution to estimation methodology is the exact Gaussian estimation method proposed by Bergstrom (1983, 1985, 1986), who pointed out the benefits of this last approach in terms of computational efficiency of estimators. The Gaussian method requires, however, the derivation of the exact discrete analogue that is induced by the continuous system.
In the last decades much work has been done on estimating continuous time models based on the exact discrete analogue. In the context of non-stationary continuous systems, Bergstrom (1997) developed an algorithm for the Gaussian estimation of a mixed first- and second-order stochastic differential equation system with mixed sampling and unobservable stochastic trends, which allows for the possibility of cointegration. However, the exact discrete analogue obtained by the author exhibits some excess differencing since it is written entirely in terms of first differences instead of lagged levels along with lagged differences as it is specified in cointegrated systems formulated directly in discrete time. As noted by Bergstrom (1997, 2009), the Gaussian likelihood remains invariant under differencing, but the discrete time representation is not adequate to investigate the sampling properties of the estimates due to the presence of unit roots in the moving average part of the discrete model. A more precise specification of the discrete analogue is, therefore, required.
Chambers (2009) proposed an alternative approach to derive the exact discrete analogue, which is based on the idea of replacing unobservable components with their observable counterpart, whose difference is then assigned to the disturbance term (see also Phillips, 1991a, who first proposed this idea). The model considered by the author is a triangular system of first-order stochastic differential equations with mixed sampling and observable stochastic trends. The resulting discrete analogue retains the triangular form of the continuous time system, and does not suffer from the excess differencing.
Many economic variables often exhibit a complex correlation structure, which requires the use of higher order differential equations. It is, therefore, relevant to extend the methods used by Chambers (2009) to more complex continuous time models, and this is the main purpose of this paper. We consider a system of mixed first- and second-order stochastic differential equations with mixed sampling and observable stochastic trends, and derive the corresponding exact discrete representation. Some formulae to implement the Gaussian estimation are also provided.
It is well known that parameter estimators in continuous time diffusion processes can suffer from substantial bias in finite samples (see for example Tang and Chen, 2009; Wang et al., 2011). The estimation bias has been widely studied in the context of stationary models, but has not received much attention in the cointegration framework. Another aim of this paper is to explore the finite sample properties of the Gaussian estimator of parameters in cointegrated continuous time systems with mixed sampling. For this purpose, we conduct a Monte Carlo experiment using two bivariate models, a system of first-order stochastic differential equations and a system of first- and second-order stochastic differential equations.
The remainder of this paper is organised as follows. Section 2 briefly describes the first- order model. In Section 3 the mixed-order system is formulated and the exact discrete representation is derived. Section 4 describes the computation of the Gaussian likelihood. Section 5 reports some simulations results to examine the finite sample properties of the Gaussian estimator and Section 6 concludes.