Forecast in the LRM

Summary

Setup:

• The LRM with random sampling and GM

\[y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+…+β_kx_{i,k}+ε_i, i=1,…,n\]

Forecast

• If we have estimated \(β_1,…,β_k\) and our estimates are \(b_1,…,b_k\) then we can forecast \(y\) for a given set of values of \(x_2…,x_k\) using

\[\hat{y}=b_1+b_2x_2+…+b_kx_k\]

• Since we are forecasting \(y\) for values of \(x\) that are not necessarily in our sample, the forecast is also called out of sample prediction . Note that the forecast for an observation in the sample is precisely the fitted value.