Splinification of AR and Applications to Yield Curve Forecast

One of the main indicators for the economical situation is the Yield Curve (YC). In particular, the most important part in the solution of the problem of the Asset liability management of a financial institution investing in bonds is the construction of the model of YC. There are many methods devoted to YC estimation, modelling and forecasting. YC is most often derived from all the bonds traded on the market, by applying different interpolation methods. There exist many approaches for the interpolation (pricing, building) of YC; a nice recent survey is available in [10]. Let us call the attention to the fact that the authors of macro-financial models (Duffie, D. and R. Kan, Duffee, G.R, Ang and Piazzesi, Rudebusch and Wu, Hohrdahl-Tristani and Vestin, Evans and Marshall) use the so-called VAR(1) models, since the last are the discrete analogue to the multidimensional Ito processes. During the last years in the financial economics area a number of strictly specific models of time series have been developed, as e.g. ARMA-GARCH and their modifications [5, 8]. These are models suitable for processes with a varying dispersion which are characterized by the so-called ”volatility clustering”. Examples for such successful models are the autoregression models AR(p), the moving average models MA(q), and their combinations ARMA(p, q), ARIMA(p, k, q), as well as the multidimensional generalizations VAR(p), cf. [8]. On the other hand, together with the classical polynomial models for approximation of the time series during the last 40 years strong influences have found the so-called spline models. Splines have been established nowadays as a lot more flexible method, compared with the usual polynomials, for interpolation and approximation of the data, since they are composed of many polynomial pieces matched smoothly. The last property implies a very adequate representation of the changing trends of the financial data [3], [9]. Splines are very useful also in the cases when we do not have enough data of the YC in some intervals of time (data gaps). They can provide a very good approximation to the real data. For the description of the spline models of interpolation (fitting, pricing) of YC, see [10], [2] and [6]. In the present study we propose the application of the spline version of Autoregression models, which we call ”splinified AR”. The model is constructed as follows: we subdivide the time interval of the observed data into smaller intervals (whose endpoints are the breakpoints for the spline) and assume that the autoregression parameters are different in every subinterval. The main point is that we choose the parameters in such a way that we have C smoothness of the autoregression equation, thus creating a kind of a ”cubic spline AR model” or ”splinified AR(2) model”. During the first step, we make calibration of the parameters of AR by means of maximizing a corresponding likelihood function. The second step is to optimize with respect to the breakpoints of the spline by seeking for such a choice of the breakpoints which would maximize the forecasting performance of our model; the