Estimating Dynamic Euler Equations with Multivariate Professional Forecasts
Smith, Gregor W., Yetman, James, Economic Inquiry
Dynamic Euler equations restrict multivariate forecasts. The aim of this paper is to study an example of these restrictions applied to professional forecasts and so introduce a new way to test these key building blocks of dynamic economic models. Economists have previously used forecast survey data in estimating and testing Euler equations in the form of asset-pricing models. For example, exchange-rate forecasts have been used in testing uncovered interest parity and measuring risk premia in the foreign exchange market. Analysts' forecasts of firm cash flows or other variables have been used to measure surprises that affect stock prices. But these studies generally study the link between the median forecast of a fundamental and an observed asset price or return. The median is adopted either because individual forecasts are not available or because some single, summary statistic must be selected for use in a statistical model.
The main innovation of this paper is to use forecasts both for the fundamentals and for the asset returns. We use only forecast data. As a result, we can use the entire cross section of individual forecasts and so add many observations to the statistical problem of estimating parameters and testing the model. This approach raises two questions. With no realized data, are we still estimating the parameters of interest? Are there gains from this approach? We answer yes to both questions. The first answer simply uses the law of iterated expectations, where we take an Euler equation and project it on the forecasters' information set (actually, the forecasters do the projecting for us).
The second answer follows from the empirical comparison of our approach both with estimation using median forecasts and with traditional estimation using realized data for the same series and time periods. In that comparison we see two benefits. Our standard errors (which allow for residual correlation across forecasters) are 25%-85% smaller than those found using estimation with median forecasts or with the traditional approach that uses instrumental-variables estimation in realized data, indicating greater precision. Additionally, our use of disaggregated data avoids aggregation bias that appears to affect some of the estimates.
Our application is to the intertemporal, consumption Euler equation that links nominal interest rates to the inflation rate and the growth rate of real consumption. This application is a natural one both because forecast data are available for these variables and because the results can thus be benchmarked against many studies using historical data. Economists have studied this relationship extensively using data on consumption (or other variables that affect marginal utility) and asset returns. Cochrane (2001) reviews theory and evidence. The simplest versions based on constant relative risk aversion (CRRA) utility often can be rejected in aggregate data. But this relationship, or some variant of it, is still a component of many dynamic, economic models.
Our study investigates whether professional forecasts embody links implied by the consumption Euler equation between macroeconomic variables and interest rates. After all, forecasters are paid to filter information and to make accurate predictions. It is interesting to see whether their forecasts implicitly link returns with inflation or with the real side of the economy. If these links held in the data, then using them to link forecasts would improve accuracy and precision.
Our first finding is that there is mixed evidence to support the Euler equation. The response of interest rates to inflation rates is forecast to be near 1, in accordance with the Fisher effect. The response of interest rates to consumption growth, while generally positive, is small and sometimes statistically insignificant, as in previous studies with realized data. Our main aim, however, is to suggest a new way of testing any asset-pricing model. …