Appendix A: Full Regression Results
Tables A.1, A.2, A.3, and A.4,show the full regression results. In each table, columns (1)-(5) report results separately for each quintile or starting credit score. For computational stability, balances, purchase volume, and wages are re-scaled to $1000s and LF and r are measured in 10 percentage points (r = 14.99% → 1.499) or dollar (LF = $39 → 3.9) units, respectively. Account age is measured in years and the trend is the number of years from September 2010. In other words, it takes value \(\frac {1}{12}\) in October 2010 and value \(-\frac {1}{12}\) in August 2010. Bootstrapped standard errors, in parentheses, are at the account level and reported based on 500 draws.
Table A.1 First Stage Regressions Table A.2 Second Stage Regressions (Payments) Table A.3 Second Stage Regressions (Borrowing) Table A.4 Second Stage Regressions (Late) Table A.1 reports first stage results.
Table A.2 reports the full (second stage) results for the payments equation.
Table A.3 reports the full (second stage) results for the borrowing equation.
Table A.4 reports the full (second stage) results for the late payment equation. Note that, since it is a binary equation, the second moment of the normal distribution, σ, is normalize to 1.
Appendix B: Pass through of treasury yields to interest rates
In this section, I provide evidence on the pass through of the cost of funds to the interest rate on accounts. As has been shown in previous work, the cost of funding accounts is closely tied to the 1-Year T-Bill rate (Ausubel 1991). As I show below, lender decisions to change existing accounts’ interest rates are associated with movements in the 1-Year T-Bill rate. Specifically, we estimate:
$$ {\varDelta} rate_{jm} = \alpha_{0} + \alpha_{1} {\varDelta}_{-1} T-Bill_{m} + \epsilon_{jm}, $$
(B.1)
where alpha gives the percentage point pass through of the T-Bill rate to the individual account’s interest rate.
Table B.1 shows pass through regressions relating the change in an account’s interest rate with a past change in the Treasury yield. Column (1) shows the average within group effect, whereby the group is defined as in Section 5.2. Columns (2)-(6) shows these effects separately by credit score quintile. Standard errors are clustered at the account level.
Table B.1 Pass Through of the Cost of the 1-Year T-Bill Rate As can be seen in the table, lenders pass through between 1 and 8 percent of the change in the 1-Year T-Bill rate to the price of existing accounts, about 6 percent on average. This is a monthly change, and does not include any effects on the interest rate from new accounts - as was shown in Ausubel (1991). While substantial, note that lenders are heterogeneous in their funding strategies, whereby not all would respond to a change in this rate. Moreover, funding heterogeneity is likely larger in this period as compared to that described in Ausubel (1991). This may then under-count the pass-through of lenders costs to the price on existing accounts. Nevertheless, it provides evidence in favor of this view.
Appendix C: Model example with quadratic utility
C.1 Baseline model
In this section, we provide an example of the model with a quadratic specification. This exposition gives a clearer illustration of the model’s mechanics by giving with closed form solutions for purchases (\(c^{\ast }_{ijm}\)), the repayment amount (\(g^{\ast }_{ijm}\)), and late payment, \(\ell ^{\ast }_{ijm}\). Suppose that
$$ \begin{array}{@{}rcl@{}} && u_{i}(c_{ijm}) = \alpha_{i} \cdot c_{ijm}-\frac{\gamma_{i}}{2} \cdot c_{ijm}^{2} \\ && k_{i}(g_{ijm}) = \theta_{i} \cdot g_{ijm} +\frac{\epsilon_{i}}{2} \cdot g_{ijm}^{2} \\ && \overline{K}_{i}[b_{ijm}] = \mathbb{E} \left[\mu_{i} (b_{ijm} + \mathbbm{1}[\cdot] LF) + \frac{\rho_{i}}{2}(b_{ijm} + \mathbbm{1}[ \cdot ] LF)^{2} |{\varOmega}_{ijm}\right], \end{array} $$
(C.1)
$$s.t. \alpha_{i},\gamma_{i}, \theta_{i}, \epsilon_{i}, \mu_{i}, \rho_{i} >0. $$
In addition, αi > γi, purchase utility shows diminishing returns, and αi > 𝜃i, the user prefers to make positive purchases on the card. The parameters also depend on the state, \({\varOmega }_{ijm}= \{b_{0}, X_{j}, y_{im},\eta ^{c}_{ijm},\eta ^{g}_{ijm}\}\), where b0 is the initial debt, Xj captures card j′s attributes, yim captures user i′s characteristics, \(\eta ^{c}_{ijm}\) is an idiosyncratic purchase shock, and \(\eta ^{g}_{ijm}\) is an idiosyncratic repayment shock.
In optimality, we have that
$$ \begin{array}{@{}rcl@{}} {}\alpha_{i} - \gamma_{i}c^{\ast}_{ijm} \!&=&\! \theta_{i}+\epsilon_{i}g^{\ast}_{ijm} \\ {}\theta_{i} +\epsilon_{i}g^{\ast}_{ijm} \!&=&\! \beta_{i}(1 + r)E\left[\mu_{i} + \rho_{i} \left( (b_{0}+c^{\ast}_{ijm} - g^{\ast}_{ijm})(1+r)+\mathbbm{1}(\cdot)LF\right) |{\varOmega}_{ijm}\right] \end{array} $$
(C.2)
Note from Eq. C.2, if the utility gains from credit card purchases decline slower in the purchase amount, e.g., when the user generates exceptional value from making purchases on their card, then trade-off in purchase, repayment trends towards purchases. The amount of unpaid balances may rise or fall dependent on whether (1) repayment today is costly, and/or (2) the future cost of debt is high. Alternatively, when a user’s marginal cost of future debt increases, ρi ↑, or when βi ↑, the user purchases less in equilibrium and repays more of her debt. Put together, this illustrates that purchases decline when the rate increases, and so does the the amount of debt held on the card.
Whether the user pays late or not depends on her equilibrium repayment amount without late payment (\(\mathbbm {1}(\cdot )=0\)) and how this compares to the minimum payment amount gmin. The probability of paying late for the account can then be calculated as:
$$ Pr(Late)_{ijm} = \Pr (\beta_{i}(1+r){\varDelta}\overline{K}_{i}|_{g^{min}}<\theta_{i}+\epsilon_{i}.g^{min}) $$
(C.3)
where
$$ {\varDelta}\overline{K}_{i}|_{g^{min}}=LF \times \mathbb{E}\left[\mu_{i}+\rho_{i}\left[\left( b_{0} + c^{\ast}_{ijm} - g^{min}\right)(1+r)+\frac{LF}{2}\right] | {\varOmega}_{ijm}\right]. $$
When LF decreases, the probability of paying late increases for a given account. Conditional on paying late, the amount of credit card purchases and the amount of unpaid balances both decrease in the level of late payment fee. Thus, the model predicts a higher late payment propensity, a higher volume of purchasing, and greater debt levels when the late fee declines.
C.2 Extended model: Focusing
We also extend the quadratic example to illustrate how the focusing theory prediction could be obtained if we change the utility function of an account. Suppose that there is a positive weight, 0 ≤ wi ≤ 1, in front of the interest rate costs of future debt The utility function of the account becomes:
$$ \begin{array}{@{}rcl@{}} && {}\alpha_{i} - \gamma_{i}c^{\ast}_{ijm} = \theta_{i}+\epsilon_{i}g^{\ast}_{ijm} \\ &&{} \theta_{i} +\epsilon_{i}g^{\ast}_{ijm} = \beta_{i}(1+r)E\left[\mu_{i}+\rho_{i} \left( w_{i}(b_{0}+c^{\ast}_{ijm} - g^{\ast}_{ijm})(1+r) + \mathbbm{1}(\cdot)LF\right) |{\varOmega}_{ijm}\right] \end{array} $$
(C.4)
Consider a decline in LF. Instead of constituting a shift in the right hand side of the optimality condition, like for the baseline case, it represents a shift in the c∗/g∗ relationship. The reduction in LF increases wi and so leads the user to internalize a larger fraction of the cost of her debt. For users borrowing more, the latter effect should dominate the former. Further, different levels of credit card borrowing are affected differently by a decline in LF. A user responds more to changes in rate and this increase in response becomes larger when she carries more credit card debt. Aggregating across users, we may find an lower unpaid balances (debt) in response to reduced LF. Moreover, this reaction is more pronounced among those with a high level of unpaid balances.
Appendix D: Robustness
In this section we provide more details of the robustness checks for our analysis. One important worry is that much of our results are colored by the effects of the Great Recession, which is at its height during our period of analysis. To assuage these concerns we identify states highly exposed to the crisis using the Economic Security Index of (Hacker et al. 2012; Hacker et al. 2014). Specifically, we compare our results in the full sample to those in which we exclude areas highly impacted by the crisis. These results are shown in Table D.1.
Table D.1 Comparison to Low Exposure States Columns (1) and (2) of Table D.1 report results on rate and late fee responses, respectively, for the full sample. Columns (3) and (4) report analogous results for the low exposure sample. As is evident in the table, our results are remarkably resilient to this pruning. This holds across the entire credit score distribution and for all three of payments, borrowing, and late payment. While there may be effects of the Great Recession no accounted for in our analysis, the results of this check indicates that they are not obviously first order to our results.
Another concern is the impact of our sample selection. Namely, our exclusion of accounts moving into non-repayment (> 60 days overdue) and new accounts created during our period of analysis. While we cannot directly add these into our framework, we instead re-create our summary statistics using the full sample to get a better understanding of how these two sample differ based on observables used in the estimation.
Table D.2 shows our summary statistics for the full population of general purpose cards in the CCDB. As compared to our estimation sample, the table shows that the broader population consists of accounts that are a bit older, with slightly higher credit scores, and that face lower interest rates. As might be expected, these differences are most apparent at the lower end of the credit score distribution. One particularly apparent difference is the higher propensity of late payment among those at the bottom quintile of the credit score distribution, 5.55 vs. 9.72 percent. Nevertheless, the broad patterns of usage established in Table 1 remain largely unchanged when including all accounts in the CCDB’s general purpose portfolio.
Table D.2 Summary Statistics for Full Population Appendix E: Estimation details
In this section we detail the procedure described in Section 5 for estimating Eqs. 9, 10, and 11. Equations 9, 10 correspond to the purchase and balance equations for B ∈{U,C}, respectively, and are a type I Tobit. Equation 11, corresponding to the late payment equation L, is a binomial Probit. For each equation, we first posit its likelihood function and respective Score (S) and Hessian (H). We then derive the analytical bias adjustment term (Hahn and Newey 2004; Arellano and Hahn 2007). Finally, we describe the iterative Efficient Newton-Raphson (ENR) procedure used for estimation (Hospido 2012) and relevant standard error calculations.
E.1 Likelihood equations
The Tobit likelihood, corresponding to the purchase volume and revolving balance Eqs. 9, 10, respectively, takes the following form:
$$ \mathscr{L}^{B}_{it} = c_{it} \cdot \left[\!-ln(\sqrt{2\pi}) - ln(\sigma) - \frac{(y - x\theta - \alpha_{i})^{2}}{2\sigma^{2}}\! \right] + (1 - c_{it}) \cdot ln\left[{\varPhi} \left( \frac{lb - x\theta - \alpha_{i}}{\sigma}\right)\right] $$
(D.1)
where cit = 𝟙 (yit > lb) and lb is the censored lower bound, in this case lb = 0. Moreover, 𝜃 is a (J − 1) × 1 vector of parameters and αi (i = 1,...,N) and σ are scalars. As shown in Olsen (1978), the above likelihood is not globally concave in the parameter space. To ensure global concavity, we transform \({\mathscr{L}}^{B}_{it}\) as follows
$$ \begin{array}{@{}rcl@{}} \mathscr{L}^{B,O}_{it} &=& c_{it} \cdot \left[-ln(\sqrt{2\pi})+ln({\varDelta}) - \frac{1}{2}({\varDelta} y-x\beta -\eta_{i})^{2}) \right] \\&&+ (1-c_{it}) \cdot ln\left[{\varPhi}(0-x\beta-\eta_{i})\right] \end{array} $$
(D.2)
where \(\beta = \frac {\theta }{\sigma }\), \(\eta _{i} = \frac {\alpha _{i}}{\sigma }\), and \({\varDelta } = \frac {1}{\sigma }\). Then we can write the Score (SB,O) and Hessian (HB,O) of the likelihood as (J + N) × 1 and (J + N) × (J + N) matrices, respectively with
$$ S^{B, O} = \left[\begin{array}{c} \frac{\partial \mathscr{L}^{B, O}}{\partial {\upbeta}_{j}} \\\\ \frac{\partial \mathscr{L}^{B, O}}{\partial {\varDelta}} \\\\ \frac{\partial \mathscr{L}^{B, O}}{\partial \eta_{i}} \end{array}\right] = \left[\begin{array}{c} {\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}} x^{j}_{it} \cdot \left[c_{it} \cdot (y-x\upbeta-\eta_{i}) - (1-c_{it}) \cdot {\varLambda}(\varphi_{it})\right] \\\\ {\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}} c_{it} \cdot \left[\frac{1}{{\varDelta}} - y_{it}({\varDelta} y_{it} - x_{it}\upbeta - \eta_{i})\right] \\\\ {\sum\limits_{t=1}^{T}} \cdot \left[c_{it} \cdot (y-x\upbeta-\eta_{i}) - (1-c_{it}) \cdot {\varLambda}(\varphi_{it})\right] \end{array}\right] $$
(D.3)
$$ H^{B,O} =\left[ \begin{array}{ccc} \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\upbeta}_{j}\partial\beta_{k}} & \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial \beta_{j}\partial{\varDelta}} & \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial \beta_{j}\partial\eta_{i}} \\\\ \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\varDelta}\partial\beta_{j}} & \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\varDelta}^{2}} & \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\varDelta} \partial \eta_{i}} \\\\ \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial \eta_{i}\partial\beta_{j}} & \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial \eta_{i} \partial {\varDelta} } & \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial \eta_{i} \partial\eta_{j}} \end{array}\right], $$
(D.4)
and
$$ \left[\begin{array}{c} \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\upbeta}_{j}\partial{\upbeta}_{k}} \\\\ \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\upbeta}_{j}\partial{\varDelta}} \\\\ \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\upbeta}_{j}\partial\eta_{i}} \\\\ \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\varDelta}^{2}} \\\\ \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\varDelta} \partial \eta_{i}} \\\\ \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial \eta_{i} \partial\eta_{j}} \end{array}\right] = \left[\begin{array}{c} {\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}} x^{j}_{it} x^{k}_{it} \cdot \left[-c_{it} - (1-c_{it}) \cdot {\varLambda}(\varphi_{it}) \cdot (\varphi_{it} + {\varLambda}(\varphi_{it}))\right]\\\\ {\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}} c_{it} \cdot x^{j}_{it} \cdot y_{it} \\\\ {\sum\limits_{t=1}^{T}} x^{j}_{it} \left[ -c_{it} - (1-c_{it}) \cdot {\varLambda}(\varphi_{it}) \cdot (\varphi_{it}+ {\varLambda}(\varphi_{it})) \right]\\\\ {\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}} - c_{it} \cdot \left( \frac{1}{{\varDelta}^{2}} + y^{2}_{it} \right) \\\\ {\sum\limits_{t=1}^{T}} c_{it} \cdot y_{it} \\\\ \mathcal{1}(i=j) \cdot {\sum\limits_{t=1}^{T}} \left[-c_{it} - (1-c_{it}) \cdot {\varLambda}(\varphi_{it}) \cdot (\varphi_{it} + {\varLambda}(\varphi_{it}))\right] \end{array}\right], $$
where φit ≡−(xitβ + ηi) and \({\varLambda }(x) \equiv \frac {\phi (x)}{\Phi (x)}\).
The Probit likelihood, corresponding to the late payment equation, takes the following form
$$ \mathscr{L}_{it}^{L} = d_{it} \cdot ln\left( {\varPhi}(x\theta+\alpha_{i})\right) + (1-d_{it}) \cdot ln\left( 1-{\varPhi}(x\theta+\alpha_{i})\right) , $$
(D.5)
where, because it is not identified, the variance of the Φ is normalized to σ = 1 and Ψit ≡ xit𝜃 + αi. The Score (SL) is given by
$$ S^{L} =\left[ \begin{array}{c} \frac{\partial \mathscr{L}^{L}}{\partial \theta_{j}} \\\\ \frac{\partial \mathscr{L}^{L}}{\partial \alpha_{i}} \end{array}\right] =\left[ \begin{array}{c} {\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}} x^{j}_{it} (d_{it} \cdot {\varLambda}({\varPsi}_{it})- (1-d_{it}) \cdot {\varLambda}(-{\varPsi}_{it})) \\\\ {\sum\limits_{t=1}^{T}} \left[d_{it} \cdot {\varLambda}({\varPsi}_{it})- (1-d_{it}) \cdot {\varLambda}(-{\varPsi}_{it})\right] \end{array}\right], $$
(D.6)
where, as above, \({\varLambda }(x) \equiv \frac {\phi (x)}{\Phi (x)}\). We write the Hessian (HL) as a (K + N) × (K + N) symmetric matrix with the following form
$$ H^{L} =\left[ \begin{array}{cc} \frac{\partial^{2}\mathscr{L}^{L}}{\partial \theta_{j}\partial\theta_{k}} & \frac{\partial^{2}\mathscr{L}^{L}}{\partial \theta_{j}\partial\alpha_{i}} \\ \frac{\partial^{2}\mathscr{L}^{L}}{\partial \alpha_{i}\partial\theta_{k}} & \frac{\partial^{2}\mathscr{L}^{L}}{\partial \alpha_{i} \partial\alpha_{j}} \end{array}\right] =\left[ \begin{array}{cc} H^{L}_{\theta\theta} & H^{L}_{\theta\alpha} \\ H^{L}_{\alpha\theta} & H^{L}_{\alpha\alpha}. \end{array}\right]. $$
(D.7)
Then let
$$ {\varGamma}_{it} = \left[ d_{it}\cdot {\varLambda}({\varPsi}_{it}) \cdot (-{\varPsi}_{it} - {\varLambda}({\varPsi}_{it})) - (1 - d_{it})\cdot {\varLambda}(-{\varPsi}_{it}) \cdot (-{\varPsi}_{it}+{\varLambda}(-{\varPsi}_{it}))\right], $$
(D.8)
and write
$$ \left[ \begin{array}{c} \frac{\partial^{2}\mathscr{L}^{L}}{\partial \theta_{j}\partial\theta_{k}} \\\\ \frac{\partial^{2}\mathscr{L}^{L}}{\partial \theta_{j}\partial\alpha_{i}} \\\\ \frac{\partial^{2}\mathscr{L}^{L}}{\partial \alpha_{i} \partial\alpha_{j}} \end{array}\right] =\left[ \begin{array}{c} {\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}} x^{j}_{it} x^{k}_{it} \cdot {\varGamma}_{it} \\\\ {\sum\limits_{t=1}^{T}} x^{j}_{it} \cdot {\varGamma}_{it} \\\\ \mathcal{1}(i=j) \cdot {\sum\limits_{t=1}^{T}} {\varGamma}_{it} \end{array}\right] $$
(D.9)
These above equations thus characterize the un-adjusted likelihoods of interest and their respective Score and Hessian expressions.
E.2 Analytically adjusted likelihood
To correct for the asymptotic bias resulting from small T and large N, the incidental parameters problem, we use an analytically derived bias correction of the concentrated likelihood as described in Hahn and Newey (2004) and Arellano and Hahn (2007). Although there are other simpler methods of bias reduction, such as automatic jackknife methods, we use the analytical method because we have only one change in the price of the late fees. As a result, we must rely more on the structure of the likelihood in our identification. Following Arellano and Hahn (2007), the asymptotic bias term of the concentrated likelihood is given by
$$ b_{i}(\theta) = \frac{1}{2} \cdot H^{-1}_{(\alpha) i}(\theta) \cdot {\Upsilon}_{(\alpha) i}(\theta) = \frac{1}{2} \cdot \mathbb{E}_{i}\left[ - \frac{\partial^{2}\mathscr{L}(\theta,\alpha(\theta))}{\partial \alpha^{2}} \right]^{-1} \cdot \mathbb{E}_{i}\left[\frac{\partial \mathscr{L}(\theta,\alpha(\theta))^{2}}{\partial \alpha}\right] $$
(D.10)
Then, following Hospido (2012), we reformulate the above expression in terms of the original, un-concentrated likelihood, as an input into the estimation. Given this approach, we write down the following estimator of the bias term
$$ \hat{b}_{i}(\theta,\alpha) = - \frac{1}{2} \cdot \left[{{\sum}_{t=1}^{T}} \frac{\partial^{2}\mathscr{L}_{it}(\theta,\alpha)}{\partial \alpha^{2}} \right]^{-1} \cdot \left[{{\sum}_{t=1}^{T}} \frac{\partial \mathscr{L}_{it}(\theta,\alpha)}{\partial \alpha}^{2} \right] $$
(D.11)
We then subtract the bias from the original likelihood to arrive at the adjusted likelihood
$$ \mathscr{L}^{A}(\theta,\alpha) = {\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}} \mathscr{L}_{it}(\theta,\alpha) - {\sum\limits_{t=1}^{T}} b_{i}(\theta,\alpha) $$
(D.12)
From above, it follows that SA = S − Sb and HA = H − Hb where Sb and Hb are the Score and Hessian of the bias term, respectively. It follows that we can write
$$ S^{b} = {\sum\limits_{i=1}^{N}}\left[ \begin{array}{c} \frac{\partial b_{i}}{\partial \theta_{j}}\\\\ \frac{\partial b_{i}}{\partial \alpha_{i}} \end{array}\right] = - \frac{1}{2} \cdot {\sum\limits_{i=1}^{N}} \left[ \begin{array}{c} H^{-1}_{(\alpha) i} \cdot \left( \frac{\partial {\Upsilon}_{(\alpha) i}}{\partial \theta} - b_{i} \cdot \frac{\partial H_{(\alpha) i}}{\partial \theta}\right)\\\\ H^{-1}_{(\alpha) i} \cdot \left( \frac{\partial {\Upsilon}_{(\alpha) i}}{\partial \alpha} - b_{i} \cdot \frac{\partial H_{(\alpha) i}}{\partial \alpha}\right) \end{array}\right] $$
(D.13)
and
$$ H^{b}= -\frac{1}{2} {\sum\limits_{i=1}^{N}} \left[ \begin{array}{cc} \frac{\partial^{2}b_{i}}{\partial \theta_{j}\partial\theta_{k}} & \frac{\partial^{2}b_{i}}{\partial \theta_{j}\partial\alpha_{i}} \\\\ \frac{\partial^{2}b_{i}}{\partial \alpha_{i}\partial\theta_{j}} & \frac{\partial^{2}b_{i}}{\partial \alpha_{i}\partial\alpha_{j}} \end{array}\right] $$
(D.14)
where
$$ \left[ \begin{array}{c} \frac{\partial^{2}b_{i}}{\partial \theta_{j}\partial\theta_{k}} \\\\ \frac{\partial^{2}b_{i}}{\partial \theta_{j}\partial\alpha_{i}} \\\\ \frac{\partial^{2}b_{i}}{\partial \alpha_{i}\partial\alpha_{j}} \end{array}\right] = \left[ \begin{array}{c} H^{-1}_{(\alpha) i} \left[\left( \frac{\partial^{2}{\Upsilon}_{(\alpha) i}}{\partial \theta_{j}\partial\theta_{k}} - b_{i} \frac{\partial^{2}H_{(\alpha) i}}{\partial \theta_{j}\partial\theta_{k}}\right) - \left( \frac{\partial b_{i}}{\partial \theta_{j}} \cdot \frac{\partial H_{(\alpha) i}}{\partial \theta_{k}} + \frac{\partial b_{i}}{\partial \theta_{k}} \cdot \frac{\partial H_{(\alpha) i}}{\partial \theta_{j}}\right)\right] \\\\ H^{-1}_{(\alpha) i} \left[\left( \frac{\partial^{2}{\Upsilon}_{(\alpha) i}}{\partial \theta_{j}\partial\alpha_{i}} - b_{i} \frac{\partial^{2}H_{(\alpha) i}}{\partial \theta_{j}\partial\alpha_{i}}\right) - \left( \frac{\partial b_{i}}{\partial \theta_{j}} \cdot \frac{\partial H_{(\alpha) i}}{\partial \alpha_{i}} + \frac{\partial b_{i}}{\partial \alpha_{i}} \cdot \frac{\partial H_{(\alpha) i}}{\partial \theta_{j}}\right)\right] \\\\ \mathcal{1}(i = j) \cdot H^{-1}_{(\alpha) i} \left[\left( \frac{\partial^{2}{\Upsilon}_{(\alpha) i}}{\partial \alpha_{i}\partial\alpha_{j}} - b_{i} \frac{\partial^{2}H_{(\alpha) i}}{\partial \alpha_{j}\partial\alpha_{i}}\right) - \left( \frac{\partial b_{i}}{\partial \alpha_{j}} \cdot \frac{\partial H_{(\alpha) i}}{\partial \alpha_{i}} + \frac{\partial b_{i}}{\partial \alpha_{i}} \cdot \frac{\partial H_{(\alpha) i}}{\partial \alpha_{j}}\right)\right] \end{array}\right] $$
(D.15)
This completes the characterization the adjusted likelihood \({\mathscr{L}}^{A}(\theta ,\alpha )\) for the Probit and Tobit likelihoods.
E. 3 Efficient Newton-Raphson (ENR)
Given the smoothness and convexity of our objective functions, we estimate the model parameters using an efficient Newton-Raphson (ENR) algorithm laid out in Hospido (2012). This method exploits the block structure of the Hessian matrix of the log likelihood function and provides significantly increased estimation speed.Footnote 37 In what follows, denote Θ = (𝜃,α). The Kth step of the ENR algorithm is
$$ {\Theta}_{[K]}= {\Theta}_{[K-1]} - \left[ \frac{\partial^{2}\mathscr{L}({\Theta}_{[K-1]})}{\partial {\Theta}\partial{\Theta}^{\prime}}\right]^{-1} \cdot \left[\frac{\partial \mathscr{L}({\Theta}_{[K-1]})}{\partial {\Theta}} \right] $$
(D.16)
Where the Score and Hessian in their block form can be expressed as follows
$$ S^{A} = \frac{\partial \mathscr{L}^{A}}{\partial {\varTheta}} =\left[ \begin{array}{c} \underbrace{S^{A}_{\theta}}_{J \times 1}\\\\ \underbrace{S^{A}_{\alpha}}_{N \times 1} \end{array}\right] \text{and} H^{A} = \frac{\partial^{2}\mathscr{L}^{A}}{\partial {\Theta}\partial{\Theta}^{\prime}} = \left[\begin{array}{cc} \underbrace{H_{\theta\theta }}_{J \times J} & \underbrace{H_{\theta\alpha }}_{J \times N} \\\\ \underbrace{H^{\prime}_{\theta \alpha}}_{N \times J} & \underbrace{H_{\alpha\alpha }}_{N \times N} \end{array}\right] $$
(D.17)
Given the block nature of HA, re-write the Kth step of the ENR algorithm in two parts as
$$ \begin{array}{@{}rcl@{}} & \theta_{[K]} = \theta_{[K-1]} - \left[H^{A}_{\theta\theta} - H^{A}_{\theta\alpha} \left( H^{A}_{\alpha\alpha}\right)^{-1} H^{A}_{\alpha\theta}\right]^{-1} \cdot \left[S^{A}_{\theta} - H^{A}_{\theta\alpha}\left( H^{A}_{\alpha\alpha}\right)^{-1}S^{A}_{\alpha}\right] \\ \\ & \alpha_{[K]} = \alpha_{[K-1]}-H^{-1}_{\alpha\alpha} \cdot \left[S^{A}_{\alpha}+H_{\alpha\theta}(\theta_{[K]}-\theta_{[K-1]})\right] \end{array} $$
(D.18)
Note that in the Tobit case, the parameter 𝜃 includes the shape parameter of the normal distribution, whereas in the Probit case it does not as the variance is normalized to 1.
E.4 Standard error calculations
The above estimator is consistent and asymptotically normal under our assumption of i.i.d errors. It follows that
$$ \sqrt{T}\left( \begin{array}{c} \hat{\theta} - \theta_{0} \\ \hat{\alpha}_{i} - \alpha_{i0} \end{array}\right) \xrightarrow{d} N\left( 0,\left( \begin{array}{cc} I_{\theta,\theta} & I_{\theta,\alpha_{i}} \\ I_{\alpha_{i},\theta} & I_{\alpha_{i},\alpha_{i}} \end{array}\right)^{-1}\right), $$
(D.19)
$$ \sqrt{T}(\hat{\theta} - \theta_{0})\xrightarrow{d} N\left( 0,\left( I_{\theta,\theta}-I_{\theta,\alpha_{i}}I^{-1}_{\alpha_{i},\alpha_{i}}I_{\alpha_{i},\theta}\right)^{-1}\right), $$
(D.20)
where
$$ \left( \begin{array}{cc} I_{\theta,\theta} & I_{\theta,\alpha_{i}} \\ I_{\alpha_{i},\theta} & I_{\alpha_{i},\alpha_{i}} \end{array}\right) = \left( \begin{array}{cc} -\mathbb{E}\left( \frac{\partial^{2}\mathscr{L}}{\partial (\theta)^{2}}\right) & -\mathbb{E}\left( \frac{\partial^{2}\mathscr{L}}{\partial \theta\partial \alpha_{i}}\right) \\ -\mathbb{E}\left( \frac{\partial^{2}\mathscr{L}}{\partial \alpha_{i}\partial\theta}\right) & -\mathbb{E}\left( \frac{\partial^{2}\mathscr{L}}{\partial (\alpha_{i})^{2}}\right) \end{array}\right). $$
(D.21)
We use sample means as consistent estimators. We can then recover confidence intervals of the original parameters using the delta method. Nevertheless, we prefer not to assume i.i.d errors. Moreover, we use the control function approach of Smith and Blundell (1986), which gives an estimated first stage parameter. To account for this additional variation, we instead calculate standard errors using a block bootstrap, where we draw (with replacement) a sample of N accounts. This also makes it possible to generate standard errors for more complex functions of the parameters, such as the average price response.