Appendix 2: Calibration of the model
Asset and liability values
The balance sheet is decomposed into the entities shown in Table 5.
We have used net asset values for most entities. This reflects approximate matching of assets and liabilities in the case of DMO and RBNZ, and for the SOEs reflects the fact that they manage their assets and liabilities as an integrated business. We have modeled the assets and liabilities of ACC and the GSF separately. The net assets of smaller entities are grouped for modeling simplicity.
We have eliminated intra-Crown cross-holdings so that values for each entity may be different from those reported by entities. For example, whilst the EQC manages about $5 billion in assets, much of this is in government bonds. This is an internal transaction for the Crown which has no bearing on the aggregate risk exposure to the Crown's net worth. Thus, in the model, the capital associated with EQC is only about $2 billion.
Values for primary revenue and expenditure are based on accrual measurement (which is only an approximation of cash flows) with growth rates determined by using the 2009 Long-term Fiscal Statement and applying a nominal annual discount rate of 10%. The choice of discount rate is largely arbitrary, although we note that this rate is consistent with estimates of the social opportunity cost of capital (New Zealand Treasury, 2008).
Expected returns and volatility
For each asset and liability class in the model, the expected return and volatility assumptions are set out in Table 14 (annual simple returns). The general approach for the different types of conventional assets and liabilities is set out in Table 12. The assumptions for the primary balance settings are discussed in more detail below.
| Entity type | Description of approach |
|---|---|
| DMO and RBNZ | The expected returns are set equal to the current yield on 5-year government bonds. Volatility is estimated based on the standard deviation of total returns on an index of New Zealand government bonds over 1996 to 2009. |
| SOEs | Expected returns are modeled to be equal to the cost of equity which is estimated to be 10% (Macquarie, 2008). Volatility is estimated based on the market volatility of equity in similar enterprises. All SOEs are assumed to have the same return characteristics, which is a modeling simplification. |
| Crown financial institutions (CFIs) | Return and volatility assumptions are estimated based on portfolio compositions. These have been adjusted to reflect that some CFIs hold some of their portfolios in the form of government bonds, holdings of which are eliminated because they are an intra-Crown transactions. |
| Government departments and Crown entities (excluding DMO, RBNZ and Crown financial institutions) | Most assets managed by government departments and Crown entities are property, plant and equipment. Expected returns and volatility have been estimated by looking at the historic return and volatility of a New Zealand commercial property index. For estimates of risk which exclude these social assets, the financial assets of government departments and Crown entities are assumed to have the same return characteristics as government bonds. |
| ACC liability and GSF liability | Expected returns are assumed to be the same as for government bonds. Volatility is assumed to be higher than for government bonds to reflect additional risks attributable to demographics and cost uncertainties. |
| Student loans | The expected return is based on an effective interest rate, which is used to impute the asset value. Volatility is assumed to be equal to that of government bonds. |
Because of the relatively large size of primary revenue and expenditure components in the model, their return and volatility parameters are very important to the model's results. The primary balance is not a conventional asset or liability and does not have ‘returns' in the conventional sense, but instead future expected revenue and spending increases are determined using the Treasury's long-term fiscal projections.
The model does not have any mean-reversion assumed and therefore the modeled volatility should be thought of as the permanent component of changes in taxes and spending. Thus what is needed is an estimate of forecast error in the structural level of primary revenue and expenditure, holding policy settings constant.
Keene and Thomson (2005) analyse tax forecasting errors by the Treasury. Their analysis focuses on the one-year-ahead forecasts. They adjust for policy change. Their key finding was that tax revenue errors had a standard deviation of 3.2%. This is likely to have some error attributable to non-permanent effects, which would overstate error. On the other hand, forecasting tax revenue over the longer term would be expected to be subject to greater uncertainty.
Treasury's projections of GDP may also be useful as an indicator. We lookedat projections for real GDP in the Budget Economic and Fiscal Updates over 2003 to 2009. The standard deviation to the annual change in projections for real GDP was 2.4% and the standard deviation in the difference to the most up-to-date projection was 3.1% (looking at projections for the 2020 fiscal year where all variables are at trend levels). Tax revenues would tend to be more volatile than GDP, so this might understate revenue volatility.
There is a literature on estimating output gaps in real time. A US study found volatility in simulated revisions to output gap estimates to be in the range of 1.5% to 3.5%, depending on the different filtering techniques (Orphanides and van Norden, 2002). A New Zealand study (Graff, 2004) estimated much smaller errors, with average absolute deviations of 0.9% for 15 quarters over 1997 to 2000. These studies are likely to represent a lower bound on the uncertainty of future tax revenues because they only analyse uncertainty in real time.
Based on this evidence (limited as it is), we chose a value for the annual volatility of 3.0% for primary revenue and 1.5% for primary expenditure. The latter is no more than an educated guess based on the intuition that volatility in expenditure, holding policy constant, is likely to be smaller than the volatility of tax revenue.
Net capital injections
We assume that SOEs pay dividends of 4.5% of their pre-dividend equity value, reflecting the average over the past eight years. Crown financial institutions do not receive capital injections, which is a simplification and reflects the temporary suspension of contributions to the NZSF. Other assets are assumed to have a net capital injection of 2% per annum, reflecting new capital expenditure and advances, but this is only an approximation.
Correlations
Correlations are estimated using historical time series of monthly returns. Sample correlations between each time series are computed with equal weight placed on each observation. A list of the datasets is shown in Table 13. The correlation coefficients used in the model are given in Table 15. The length of the time series is relatively short – around 40 quarters.
In some cases, the same dataset is used to proxy the returns of more than one asset or liability class in the model. This has two consequences. First, where two classes have the same dataset used to proxy returns, we have had to use our judgement for the correlation between them. We did this based on what we considered to be economically plausible relationships. The second consequence of this approach is that there is no guarantee of generating a valid correlation matrix (which must have the mathematical property of being positive semi-definite). In order to generate a positive semi-definite correlation matrix, we used an algorithm developed in Higham (2002), which finds the closest valid correlation matrix to the ‘invalid' one. This algorithm finds a valid correlation matrix which for practical purposes is very close to the original specification.
A critical correlation coefficient is that between primary revenue and primary expenditure. No reliable data was available because of the endogeneity to policy choice. There is suggestive evidence in the Treasury's Long-term Fiscal Model. Here, primary revenue and expenditure are positively correlated because higher economy-wide labour productivity affects both tax revenue and expenditures on New Zealand Superannuation payments and the public-sector wage bill of existing programmes. In that model, a 1% change in tax revenues due to labour productivity variability is associated with a 0.6% change in primary expenditures, holding policy constant. Other factors can effect affect tax and not spending (or vice versa), such as changes in labour participation. We chose as a correlation parameter the value of 0.5.
