To construct the ‘yield-only’ and ‘yield-macro’ models, we use nominal government spot interest rates extracted from the conventional gilt market, real spot interest rates and implied inflation rates extracted from the index-linked gilt market by the Bank of England (2010). We use all available maturities i.e. 50 different maturities for nominal rates (starting from 6 month and ending with 25 years) and 46 maturities for real rates and implied inflation (starting from 2.5 years and ending with 25 years). As for the macroeconomic variables we use realised inflation obtained from the Retail Price Index and output gap provided by the OECD Economic Outlook publications.
The output gap, as defined by the OECD in the Economic Outlook, is the difference between actual Gross Domestic Product (GDP) and potential GDP as a percent of po- tential GDP. Potential GDP has been defined as the level of output that an economy
2The expectations hypothesis of the term structure states that movements in long rates are due to
can produce at a constant inflation rate. However an economy can temporarily produce more than its potential level of output at the cost of creating inflationary pressures. Therefore, while GDP is compiled according to international guidelines and observed the same cannot be said for the potential GDP. Not only is the methodology for es- timating potential GDP open to discussion with the estimate itself usually depending on the estimate of capital stock, the potential labour force (which in turn depends on the demographic factors and on the participation rates), the estimate for NAIRU (non-accelerating inflation rate of unemployment or structural rate of unemployment) and the level of labour efficiency (Tosetto, 2008).
The output gap is linked to the concepts of ‘capacity’ and ‘demand/supply’. When actual output exceeds the economy’s potential, the output gap is positive and when actual output is below potential output, the output gap is negative. A positive output gap is also referred to as excess demand, while a negative to as excess supply. Therefore in theory when spending in the economy is high in relation to capacity (positive output gap), this tends to put upward pressure on prices and, accordingly inflation will also tend to rise.
The output gap is often subject to considerable revision over time. This is due to the fact that as for any measure of the business cycle potential activity, which is, in this case potential output or potential GDP as a target variable is unobservable. So the measure of the gap between actual and potential output: is not well defined, sensitive to the choice of the estimation technique, and also sensitive to the available dataset and therefore itself often subject to considerable revision over time. However uncertainty about the size and the movements of the output gap is not the only one which policymakers have to face and it does not imply that the output gap and the potential output estimates are not useful, because they still contain information, even if measured with error (Tosetto, 2008).
As for the realised inflation, we calculate the annual inflation by taking the difference of the logged values of quarterly RPI data.
Chapter 5
Modelling the UK Term Structures:
The Yield-Only Model
5.1
Introduction
We use monthly data to construct the UK ‘yield-only’ model. First we introduce the data by presenting some descriptive statistics in Section 5.2. Section 5.3 discuses the ‘yield-only’ model along with the principal component analysis applied on the data, auto- and cross-correlations among the PCs, suitable models for each variable and an analysis of the residuals respectively. Section 5.4 describes how we derive the term structures back and examine the one-month ahead forecasts by constructing 95% confidence intervals for the forecasts. Furthermore, we check whether our one-month ahead forecasts satisfy the Fisher relation and whether we can forecast one of the yield curves using the other two in Section 5.5. Finally, Section 5.6 concludes.
5.2
Data
To construct the ‘yield-only’ model, we use monthly UK nominal government spot interest rates extracted from the conventional gilt market, monthly real spot interest rates and monthly implied inflation rates extracted from the index-linked gilt market by the Bank of England. As we have discussed in Chapter 2, first we fit the Cairns
model in order to use all available maturities, i.e. 50 different maturities for nominal rates (starting from 6 month and ending with 25 years) and 46 maturities for real rates and implied inflation (starting from 2.5 years and ending with 25 years).
In Table 5.1, we present the summary statistics for the fitted monthly nominal and real interest rates and implied inflation rates at representative maturities (in years). Although a typical yield curve is upward sloping, and the long rates are less volatile and more persistent than short rates, due to having a relatively short period of data we see that the means of the yield curves for different maturities are quite close to each other. Considering the standard deviations, although they do not change significantly, the volatilities decrease for nominal and implied inflation data as the maturities get longer. The minimum (maximum) values for the shortest maturities for all three yield curves are lower (higher) than the minimum (maximum) values for the longest maturities. The autocorrelation functions indicate significant correlations for one month, six months and twelve months (one year) lags in the yield curves. These high correlations show that the interest rates and implied inflation rates depend highly on their previous values. Although the autocorrelation functions decay very slowly for the three yield curves, which might indicate non-stationarity, we will assume that they are stationary. It is more an economic assumption rather than a statistical one. We do not have a sufficiently long period of data here to justify the stationarity of the yield curves, but observation over far longer periods shows that yields must be stationary (Homer, 1963).