## Non Linearity in the Sensitivity of the Mortgage Rate to the LTV

In Figure 3 we plotted the sensitivity of the fixed-rate and 2/28 hybrid mortgage rates to the first-lien LTV ratio. The sensitivity is defined as the regression coefficient on the first-lien LTV (scaled by the standard deviation) in a regression with the mortgage rate as dependent variable and the first-lien LTV, the second-lien LTV, and the other loan and borrower characteristics listed in Subsection 4.2, as independent variables.

In this subsection we study the robustness of this result to adding the square of the first-lien LTV and the square of the second-lien LTV as independent variables, therefore allowing for a non-linear functional form. In Figure 11 we report the resulting scaled marginal effect of the first-lien LTV for fixed-rate and 2/28 hybrid mortgages evaluated at a first-lien LTV of 80 percent (left panel) and 90 percent (right panel). Without non-linear terms the marginal effect is simply given by the regression coefficient. This is what we plotted in Figure 3. With the quadratic terms, the marginal effect is given by (3LTV + 2(ltv2X, where the (3s are the regression coefficients and X is the first-lien LTV ratio at which the marginal effect is evaluated.

Figure 11: Sensitivity of Mortgage Rate to First-Lien LTV Ratio Allowing for Non-Linearity

The figure shows the scaled marginal effect of the first-lien loan-to-value (LTV) ratio on the mortgage rate for first-lien fixed-rate and 2/28 hybrid mortgages, evaluated at a first-lien LTV of 80% (left panel) and 90% (right panel). The effect is determined using an OLS regression with the interest rate as dependent variable and the FICO score, first-lien LTV (and the square), second-lien LTV (and the square), debt-to-income ratio, missing debt-to-income ratio dummy, cash-out refinancing dummy, owner-occupation dummy, prepayment penalty dummy, origination amount, term of the mortgage, prepayment term, and margin as independent variables.

Scaled Marginal Effect of First-Lien LTV = 80% (%)

Scaled Marginal Effect of First-Lien LTV = 90% (%)

Scaled Marginal Effect of First-Lien LTV = 80% (%)

- FRM |
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2/28 Hybrid |
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2001 2002 2003 2004 2005 2006 2007 Year

2001 2002 2003 2004 2005 2006 2007 Year

Scaled Marginal Effect of First-Lien LTV = 90% (%)

As shown in Figure 11, the marginal effect is rising over time, consistent with the baseline case results presented in Figure 3. Moreover, we find that there is a statistically and economically significant nonlinear effect of the first-lien LTV on the mortgage rate. Comparing the left and right panels in Figure 11, the higher the first-lien LTV ratio, the more sensitive is the mortgage rate to changes in the first-lien LTV. The largest difference between the results based on specifications with and without non-linearity is observed for 2/28 hybrid mortgages in 2007 at a first-lien LTV of 90 percent (right panel). The scaled marginal effect increases by 27 basis points over the course of 3 months in 2007 when a model allows for non-linearity. In contrast, ignoring the non-linearity, as in Figure 3, the increase in the scaled marginal effect is only 13 basis points.

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