Econometrics
Measures of risk

The 1970s brought a new awareness that variations in the perceived riskiness of activities could be an important influence on economic decisions. In the financial press, for example, there has been much discussion of the effects of exchange rate volatility on trade flows. This awareness has encouraged new analyses of the quantitative importance of risk. The central problem encountered in such work has been the construction of 'proxies' to represent the level of risk. Sometimes researchers have been content with an indicator of whether risk has increased or decreased, but such measures may not be adequate for regression analyses or forecasting.

In Discussion Paper No. 127, Aman Ullah and Research Fellow Adrian Pagan discuss the problems involved in estimating regression equations in which a risk measure appears as one of the explanatory variables. They also consider techniques for estimating equations in which the variable to be explained is the level of observed risk.

A variety of proxies for risk have been proposed. One very popular measure has been a moving average of the squared deviations of a variable from its trend - a moving variance. Other proxies for inflation risk have included the variance of relative price changes across commodity groups and the variance of anticipated inflation rates over survey respondents. Pagan and Ullah find that there are difficulties with each of these when used in a regression, but they conclude that the variance of relative price movements is perhaps the best proxy.

Economic theory suggests that risk should be measured as a function of the (conditional) mean, variance and other moments of a probability distribution: a proxy will differ from this function in a random fashion. Such proxies suffer from a form of the 'errors in variables' problem, and their use in regression equations can lead to a substantial underestimate of the effect of risk on decisions. It is therefore important to devise estimators that remove this bias, and the authors argue that the most appropriate technique involves the use of instrumental variables. These are variables correlated with the true level of risk but uncorrelated with the measurement error in the proxy; they can be used to 'purge' the estimator of the effects of the measurement error.

Pagan and Ullah argue that risk must be defined in relation to some information set; if perfect predictions could be made, risk would be absent. How might such instrumental variables be chosen? The impact of risk cannot be analysed, therefore, without first specifying this information set. After this has been done, it follows that the appropriate instrumental variables can be constructed as functions of that information set. Pagan and Ullah apply this approach to models of exchange rates, interest rates, and the impact of price uncertainty on real variables in the US economy.
For some purposes only a pictorial measure of the level of risk is needed. Since the level of risk is the (conditional) variance of a variable (say inflation) the authors suggest the use of 'non-parametric' estimation techniques to estimate this variance. These techniques require fewer assumptions concerning the shape of the underlying probability distribution of the variable. Applying this method to the Canadian/US exchange rate indicated a dramatic rise in risk around the time of the election of a provincial government that advocated independence for Quebec.


The Econometric Analysis of Risk Terms
Adrian Pagan and Aman Ullah

Discussion Paper No. 127, September 1986 (ATE)