Regression Model for Exposure Estimation

Adler and Dumas defined exposure elasticity as the changes in the market value of a firm with respect to the change in exchange using following time series regression:

Rit = b0i + b1iRst + eit,     t = 1, …,T,        (1)

where Rit is the stock return for firm i, Rst is the percentage change in an exchange rate variable, measured as home currency price of foreign currency. Therefore a positive value for Rst indicates a home currency depreciation. bit is the elasticity of firm to the changes in exchange rate. This elasticity indicates the firm’s average exposure over the estimation period. b0i reflects common stock value, when expected rate of change in the exchange rate is constant over time
In order to control macroeconomic influences on returns, most recent empirical studies include market return in the model. This market return parameter not just helps to control macroeconomic influences but also reduces residual variances of the equation (1). Therefore most common form of regression equation for exchange rate exposure is described as follows:

Rit = b0i + b1iRst + b1iRmt + eit,    i =1, … , N;       t = 1, … ,T        (2)

where Rmt is the rate of return on market index, N is the number of cross sections, T is the length of the time series for each cross section.

Whether they are simple or multiple regressions each coefficient tells us what relation that input variable has with output variable (where all other input variables are held constant). In linear models in order to get the real elasticity one must multiply the slop coefficient by its input variable and then divide by the output.

The relation between inputs and output may not always be linear. We can take Cobb-Douglas production function as an example, which is widely used in empirical researches. It is defined as:

       (3)

where b1 is a constant and can be thought a scaling factor and b2 and b3 are coefficients. Although this equation is non-linear, it can be transformed to linear by taking natural logarithms of both sides.

ln yi = ln b1 + b2 lnx2i + b3 lnx3i + ui         (4)

The equation (4) is linear in the parameters but non linear in the variables. In this equation the slope is no more constant and it changes according to input and output variables. One big advantage of this equation is that the coefficients gives us partial elasticities. E.g. b2 is the partial elasticity of output yi with respect to the x2i, holding x3i constant.