Learning and the Law of Iterated Projections

Equilibrium prices or quantities in very broad classes of models depend on iterated expectations of an autoregressive forcing variable. Examples of this dependence include that of stock prices on autoregressive dividends, that of the price level or exchange rate on an autoregressive money supply, that of consumption on autoregressive income, and that of investment on autoregressive interest rates, output demand, or technology shocks. In this paper, we show that, under learning, these iterated expectations are not a certainty equivalence. Further, we show that the certainty equivalent form of iterated expectations typically used in the literature ignores covariance terms that help explain empirical anomalies in a variety of macroeconomic models. We illustrate the consequences for two examples. In the first, a permanent-income/life-cycle model of consumption, we show that learning about an autoregressive income process can explain the widely observed negative correlation of consumption growth and current income. In the second, a present-value model of an asset price, learning about the autoregressive dividend process can help explain the empirical finding that dividend yields predict excess returns. Since, under learning, the iterated expectations terms are extremely complex, we develop a numerical (Monte Carlo) method for finding a polynomial approximation. A unique feature of our method is that we approximate the agents' subjective expectations. Therefore, in the spirit of individual learning, our method utilizes only information that is available to the agent.