Indirect least squares (ILS) is a statistical method used to estimate the parameters of a model when there is a non-linear relationship between the dependent and independent variables. This method is commonly used in econometric and finance applications where the true relationship between the variables is unknown or difficult to directly measure.

One example of the use of ILS is in the estimation of the demand for a product. In this case, the dependent variable is the quantity of the product demanded, and the independent variable is the price of the product. The true relationship between these two variables is non-linear, with a negative relationship between price and quantity demanded.

Using ILS, the analyst would first estimate the demand for the product at different prices, and then use these estimates to calculate the parameters of the demand curve. This can be done using a mathematical model, such as a quadratic or polynomial model, which describes the relationship between the dependent and independent variables.

Another example of the use of ILS is in the estimation of the return on an investment. In this case, the dependent variable is the return on the investment, and the independent variable is the risk associated with the investment. The true relationship between these two variables is non-linear, with a positive relationship between risk and return.

Using ILS, the analyst would first estimate the return on the investment at different levels of risk, and then use these estimates to calculate the parameters of the return curve. This can be done using a mathematical model, such as a linear or exponential model, which describes the relationship between the dependent and independent variables.

Overall, ILS is a useful tool for estimating the parameters of a model when the true relationship between the dependent and independent variables is non-linear and difficult to directly measure. It allows the analyst to accurately estimate the relationship between the variables and make more informed decisions about the investment or product in question.