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Harry W.

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Polynomials

Assume that we have K variables X_{1},...,X_{K} and that we suspect that one of them (could be more than one), say X_{l}, may have a non linear effect on the dependent variable Y.

We can account for this nonlinearity by adding powers of X_{l} to the set of regressors:

Y = b_{0} + Σ_{k=1, k≠l}^{K}b_{k}X_{k} + b_{l,1}X_{l} + b_{l,2}X_{l}^{2} + ... + b_{l,r}X_{l}^{r}.

In practice, we start with a relatively high order (value of r), say 4, estimate the parameters and then conduct a t-test of H_{0}: b_{l}^{r} = 0. If we cannot reject H_{0}: b_{l}^{r} = 0, we drop X_{l}^{r} from the regressors and run a ren regression with a polynomial of order r - 1 and conduct a test of H_{0}: b_{l}^{r-1} = 0, and so on...

We should not let R^{2} drive our specification. Adding powers of X_{l} will mechanically increase R^{2}.

Interaction Effects

Assume that we have two regressors, for instance age (X_{1}) and gender (X_{2} = 1 if male and 0 if female). The dependent variable Y is the wage.

If we regress Y = b_{0} + b_{1}X_{1} + b_{2}X_{2} + U, we are assuming that the returns to experience are the same for men and women (and equal to b_{1}). This may be a strong assumption, wages may increase more of less quickly with age for men than for women.

To allow for this, we may interact the age and gender variables and then estimate the following linear model by OLS:

Y = b_{0} + b_{1}X_{1} + b_{2}X_{2} + b_{12}X_{1}X_{2} + U

we then have (assuming exogeneity):

∂E(Y | X_{1 }= x_{1}, X_{2} = x_{2} ) / ∂x_{1} = b_{1} + b_{12}x_{2}.

the return to age now depends on gender (on x_{2}),

Log Transformations

If a regressor is in log, its associated coefficients is not affected by a scale (or unit) change. The constant however, will be affected:

y = b_{0} + bln(αx) = b_{0} + bln(α) + bln(x).

If both regressors X and the dependent variable Y are in logs, the regression coefficient can be interpreted as the elasticity of Y w.r.t. X:

ln(y) = b_{0} + bln(x) ⇒ b = dln(y) / dln(x) = (dy/y) / (dx/x).

If Y is in logs but not X, the regression coefficient b can be interpreted as the % change in Y following a one-unit change in X.

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