This method is presented in Evans and Honkapohja (2001)
I don't understand the formula used by the least squares learning technique to form expectations in economic models. This formula is given by, in matrix notation:
delta = ((Xt)' Xt))^-1 (Xt)' Yt
where delta is estimate of the coefficients of the perceived law of motion in the model
yt is a vector of endogeneous variables xt is the history up to moment t of the exogenous variables (that could include past values of yt)
Can someone explain me what this formula does, intuitively speaking? I guess it minimizes the squared errors of the expectations vs. the actual realisations?
So delta is basically based on all information available until time t-1, and these coefficients are used in an expectation equation to forecast one (endogeneous) variable in the model, e.g. the clearing price.
You find some more background below if necessary, from: McCandless (2008):
1 Introduction
An increasingly standard way of putting bounded rationality into a model is through the use of least squares learning for the expectation process. Instead of the rational expectations that we have been using, least squares learning implies that the agents inside the model have to use data produced by the model to make least squares forecasts of expectational variables and that these forecasts are updated each period as a new data point comes available. For the reader who is uncomfortable with rational expectaions as a working paradyme, least squares forecasting provides an attractive alternative. For those who are comfortable with rational expectations, under a range of conditions, equilibria with least squares forecasting converge to rational expectations equilibria. In each period, all agents in the model use the same least squares forecasts for predicting expectational variables in the economy (the model). These forecasts are done inside the linear model of the economy and result in current values for the economy that are consistent with these projections. The data that results from this period provides another set of data points and this data is then used to update the coe¢ cients of the least squares forecasting equations. The new coe¢ cients are put into the model to solve for the next period. It may be the cse that the modeller believes that the economy has undergone structural changes or changes in policies so that the parameters of the least squares forecasting problem are time dependent. In order to allow these parameters to adjust, it is possible to add a "forgetting" factor to the updating equation. This forgetting factor gives older data less weight in the estimation process so that the values of the parameters are weighted more by what has happened recently. Adding a forgetting factor tends to make the coe¢ cients of the least squares forecasting equation vary with time and to be more responsive to recent outcomes of the economy. Recursive least squares updating is a restricted form of the Kalman Ölter. A simple demonstration of this can be found in Branch and Evans [1] or in Sargent [6], Chapter 8, Appendix A.
2 Recursive least squares updating of coe¢ cients
The Örst step is to present the recursive least squares updating procedure. This recursive least squares updating procedure is used along with a linear model that includes a expectations equations whose parameters are re -estimated each period from the updating procedure. In each period t, the linear model is run and is solved for the equilibrium values for the variables of the model for period t. The time t values of the variables are then used with the updating procedure to get new estimates for the parameters for the forecasting equations in the linear model. These new parameters are then put into the linear model and are used for forecasting in period t + 1. This process is repeated. There are two versions of the recursive ordingary least squares presented in this section. The Örst version presented is equivalent to ordinary least squares and the later is equivalent to least squares with forgetting. In the version with forgetting, older data in the linear regression is given an exponentially smaller weight for determining current coe¢ cients of the forecasting equations. Consider a simple linear model of the form yt = xt't + "t; where yt is a vector of endogenous variables, xt is the history up to moment t of the exogenous variables (that could include past values of yt) that are being used for the estimation, 't is the estimate of the coe¢ cients of the model using data up to time t 1 and "t is a vector of error terms. DeÖne Xt = [xt; xt1; :::; x0] 0 and Yt = [yt; yt1; :::; y0] 0 as the available data on xt and yt in period t and where x0 and y0 are the data for the available initial period. Ordinary least squares of the estimate of the coe¢ cients, 't , using the data available in period t is 't = (X0 tXt) 1 X0 t