Wilkie models

See also: Time series introduction,Time series modeling in finance, Time series in ModelRisk, Vose Wilkie Models

Introduction

The Wilkie Model is a stochastic asset model developed by A.D. Wilkie, which models the behavior of various economic series over time. The model is a system of linear time series models based on the methods of Box and Jenkins (1978).  The model has come to be widely used in actuarial work and is hence a benchmark for future development.

To learn how to use Wilkie models with ModelRisk see the Vose Wilkie Models topic.

The Wilkie model has a cascade structure for the investment series, as shown in the figure below:

Price Inflation can be seen as the driving force for the  other  series. The relationships between the variables are based on a blend  of  statistical  evidence  and economic  beliefs.

Below is an overview of the main Wilkie models that can be found in ModelRisk .

Price Inflation

The price inflation model is of the form:

Q(t) = Q(t-1) * exp(I(t))

where Q(t) is the price index

and:

I(t) = ln Q(t) - ln Q(t-1)

is the force of inflation over the year (t-1,t)

The model assumes that price inflation depends only on past values of itself.

As mentioned above, the price inflation model is needed in all other models.

Wage Inflation

The wages index W(t) is of the form:

W(t) = W(t-1) * exp(J(t))

so that:

J(t) = ln W(t) - ln W(t-1)

is the force of wage inflation over the year (t-1,t).

J(t) is a function of I(t) and I(t-1)

Share Yields

The share dividend yield model is of the form:

ln Y(t) = YW * I(t) + YN(t)

where YW is an inflation factor, I(t) is the price inflation and YN(t) is a first order autoregressive process.

Dividends

Wilkie's dividend yield model is of the form:

K(t) = ln D(t) - ln D(t-1)

where D(t) is the dividend index at time t and where K(t) is a function of YN(t) (see share yields).

Long Term Interest Rate (Consols Yield)

The consols yield model is of the form:

C(t) = CW * CM(t) + CN(t)

where CW is an inflation factor, CM(t) is an exponentially weighted moving average of inflation and CN(t) is a third order autoregressive process.

Short Term Interest Rate (Bank Rate)

Wilkie's bank rate model is of the form:

B(t) = C(t) * exp(-BD(t))

where C(t) is the long term interest rate at time t.

Read on: Thiel inequality coefficient