RDP 2016-11: Identifying Interbank Loans from Payments Data Appendix D: Derivation of the Model
December 2016
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For simplicity, this appendix assumes that there are no rollovers and that there is no persistence in u_{i,t}. The intuition when rollovers and persistence are included is the same. Suppose the true data-generating process is (where γ_{i,t} is the ‘net share’ in session i of day t):
With the net share varying over time, u_{i,t} consists of loan-identification errors only. However, suppose we estimate:
Then, by combining Equations (D1) and (D2):
Estimating a constant net share causes the conditional variance of (conditional on nr_{t}, and to differ from the variance that does not condition on nr_{t}:
Assuming changes in γ_{i,t} are independent of both nr_{t} and u_{i,t}:
Equation (D4) justifies the functional form we assume for the conditional heteroskedasticity in our model.
Now suppose the parameters are estimated using ordinary least squares (OLS). By construction, the estimated sample covariance between nr_{t} and will be zero. Assuming the conditions required for the weak law of large numbers hold, the sample covariance converges in probability to the population covariance. Therefore, the following holds for the population covariance:
Which, using Equation (D3), can be rewritten as:
If γ_{i,t} and nr_{t} are independent, then the population covariance can be expanded to:
Using Equations (D5) and (D6), and re-arranging, gives the following limit value for (i.e. as T → ∞, where T is the sample size):
So, in the limit, the OLS estimator of the constant net share () will equal the mean of the net share plus a bias term caused by any correlation between non-rolled loans and the loan-identification error.
If the bias term is non-zero, then an estimator for that accounts for the conditional heteroskedasticity of (such as the Gaussian maximum likelihood estimator (MLE) used in Section 5.3) will, in general, converge to a different point than the OLS estimator. However, it can be shown that the point of convergence of the Gaussian MLE will have the same form as the OLS estimator; being equal to the mean of the net share plus a bias term:
where f(nr_{t}) is the limiting conditional variance of . This bias term equals the bias term for the OLS estimator if f(nr_{t}) is a constant. As with the OLS bias term, this bias term will equal zero if nr_{t} and u_{i,t} are uncorrelated.