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Difference-in-difference analysis

2.2 Empirical issues on securitization

2.2.5 Difference-in-difference analysis

Apart from the empirical strategies above, another popular estimation method is called the Difference-in-Difference analysis. The simplest set up is one where outcomes are observed for two groups for two periods. One of the groups is exposed to a treatment in the second period but not in the first period. The second group is not exposed to the treatment during either period. In the case where the same units within a group are observed in each time period, the average gain in the second (control) group is substracted from the average gain in the first (treatment) group. This removes biases in second period comparisons between the treatment and control group that could be the result from permanent differences between those groups, as well as biases from comparisons over time in the treatment group that could be the result of trends. To understand the Difference- in-Difference analysis, it would be better to start from the basic fixed-effects model. In the fixed effects models, if a researcher is interested whether π’€π’Šπ’• is

affected by 𝑫𝑖𝑑 which is assumed to be randomly assigned. There are also time

varying covariates 𝑿𝑖𝑑 and unobserved but fixed confounders π‘¨π’Š. Therefore,

𝑬[π’€πŸŽπ’Šπ’•|π‘¨π’Š, 𝑿𝑖𝑑, 𝒕] = 𝜢 + 𝝀𝒕+ π‘¨π’Šβ€²πœΈ + π‘Ώπ’Šπ’•β€² 𝜷. (2.52) Assuming that the causal effect of individuals is additive and constant so the following equation is also true:

𝑬[π’€πŸπ’Šπ’•|π‘¨π’Š, 𝑿𝑖𝑑, 𝒕] = 𝑬[π’€πŸŽπ’Šπ’•|π‘¨π’Š, 𝑿𝑖𝑑, 𝒕] + 𝝆 (2.53) Taken together, we will have:

𝑬[π’€πŸπ’Šπ’•|π‘¨π’Š, 𝑿𝑖𝑑, 𝒕] = 𝜢 + 𝝀𝒕+ 𝝆𝑫𝑖𝑑+ π‘¨π’Šβ€²πœΈ + π‘Ώπ’Šπ’•β€² 𝜷 (2.54) This equation implies the following regression equation:

π’€π’Šπ’• = πœΆπ’Š+ 𝝀𝒕+ 𝝆𝑫𝑖𝑑+ π‘Ώπ’Šπ’•β€² 𝜷 + πœΊπ’Šπ’• (2.55)

Where πœΊπ’Šπ’•= π’€πŸŽπ’Šπ’•βˆ’ 𝑬[π’€πŸŽπ’Šπ’•|π‘¨π’Š, 𝑿𝑖𝑑, 𝒕], and πœΆπ’Š = 𝜢 + π‘¨π’Šβ€²πœΈ.

Suppose we simply estimate this model with OLS without fixed effects, then the estimation is:

π’€π’Šπ’•= 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 + 𝝀𝒕+ 𝝆𝑫𝑖𝑑+ π‘Ώπ’Šπ’•β€² 𝜷 + πœΆπ’Š+ πœΊπ’Šπ’• (2.56)

As πœΆπ’Š is correlated with the individual status 𝑫𝑖𝑑, there is a correlation of

𝑫𝑖𝑑 with error term. This will lead to biased OLS estimations. A fixed effect model

would address this problem because πœΆπ’Š would be included in the regression. 𝑫𝑖𝑑

with error term would therefore be uncorrelated and the regression would obtain

an unbiased estimator 𝝆.

In practice, there are two ways of estimating the fixed effects model: i) demeaning, or the within estimator, and ii) first differencing. With demeaning we should first calculate individual averages of the dependent variable and all explanatory variables. The we should substract the averages from the regression to obtain:

π’€π’Šπ’•βˆ’ π’€Μ…π’Š= π€π’•βˆ’ 𝝀̅ + 𝝆(𝑫𝑖𝑑 βˆ’ π‘«Μ…π’Š) + (π‘Ώπ‘–π‘‘βˆ’ π‘ΏΜ…π’Š)β€²πœ· + πœΆπ’Š+ (πœΊπ’Šπ’•βˆ’ πœΊΜ…π’Š) (2.57)

Thus πœΆπ’Š drops out and therefore the error and the regressor would no longer

be correlated.

In the first differencing way, we can also get rid of the πœΆπ’Š by:

πš«π’€π’Šπ’• = πš«π€π’• + π†πš«π‘«π‘–π‘‘+ πš«π‘Ώπ’Šπ’•β€² 𝜷 + πš«πœΊπ’Šπ’• (2.58) The Difference-in-Difference method is first introduced by Card and Krueger (1994) who analyse the effect of a minimum wage increase in New Jersey. Taken securitization as an example, we can obtain a bank securitizes loans or not. We can only observe one situation or the other, that is, at a time point, a bank can only be a securitizer or a non-securitizer, but cannot be both.

If we assume that: 1) 𝒀1𝑖𝑠𝑑 is the performance indicator of bank π’Š which has

securitized assets at state 𝒔 and time 𝒕, and 2) 𝒀0𝑖𝑠𝑑 is the performance indicator

of bank π’Š which does not have securitized assets at state 𝒔 and time 𝒕. We the assume that:

𝑬[π’€πŸŽπ’Šπ’”π’•|𝒔, 𝒕] = πœΈπ’” + 𝝀𝒕 (2.59)

In the absence of the securitization activities, a bank’s performance is

determined by the sum of a time-invariant state effect πœΈπ’” and a time effect 𝝀𝒕.

Let 𝑫𝑠𝑑 be a dummy for securitized banks after an endogenous shock, e.g., the

bankruptcy of Lehman Brothers in 2008. Assuming 𝑬[π’€πŸπ’Šπ’”π’•βˆ’ π’€πŸŽπ’Šπ’”π’•|𝒔, 𝒕] = 𝜹 is the treatment effect, observed bank performance thus can be written as:

π’€π’Šπ’”π’• = πœΈπ’” + 𝝀𝒕+ πœΉπ‘«π‘ π‘‘+ πœΊπ’Šπ’”π’• (2.60) For example, for banks with securitization activities, the performance before the bankruptcy of Lehman Brothers in 2008 is:

𝑬[π’€π’Šπ’”π’•|𝒔 = π’”π’†π’„π’–π’“π’Šπ’•π’Šπ’›π’†π’“, 𝒕 = 𝒃𝒆𝒇𝒐𝒓𝒆 πŸπŸŽπŸŽπŸ–] = πœΈπ’”π’†π’„π’–π’“π’Šπ’•π’Šπ’›π’†π’“+ π€π’‘π’“π’†πŸŽπŸ– (2.61)

And the performance after the bankruptcy of Lehman Brothers in 2008 is:

𝑬[π’€π’Šπ’”π’•|𝒔 = π’”π’†π’„π’–π’“π’Šπ’•π’Šπ’›π’†π’“, 𝒕 = 𝒂𝒇𝒕𝒆𝒓 πŸπŸŽπŸŽπŸ–] = πœΈπ’”π’†π’„π’–π’“π’Šπ’•π’Šπ’›π’†π’“+ π€π’‘π’π’”π’•πŸŽπŸ– (2.62)

Therefore, the difference between the securitizers’ performance before

and after 2008 is:

𝑬[π’€π’Šπ’”π’•|𝒔 = π’”π’†π’„π’–π’“π’Šπ’•π’Šπ’›π’†π’“, 𝒕 = 𝒂𝒇𝒕𝒆𝒓 πŸπŸŽπŸŽπŸ–] βˆ’

𝑬[π’€π’Šπ’”π’•|𝒔 = π’”π’†π’„π’–π’“π’Šπ’•π’Šπ’›π’†π’“, 𝒕 = 𝒃𝒆𝒇𝒐𝒓𝒆 πŸπŸŽπŸŽπŸ–] = π€π’‘π’π’”π’•πŸŽπŸ–βˆ’ π€π’‘π’“π’†πŸŽπŸ–+ 𝜹 (2.63)

Similarly, for non-securitized banks, the performance before the bankruptcy of Lehman Brothers in 2008 is:

𝑬[π’€π’Šπ’”π’•|𝒔 = π’π’π’π’”π’†π’„π’–π’“π’Šπ’•π’Šπ’›π’†π’“, 𝒕 = 𝒃𝒆𝒇𝒐𝒓𝒆 πŸπŸŽπŸŽπŸ–] = πœΈπ’π’π’π’”π’†π’„π’–π’“π’Šπ’•π’Šπ’›π’†π’“+ π€π’‘π’“π’†πŸŽπŸ– (2.64) And the performance after the bankruptcy of Lehman Brothers in 2008 is:

𝑬[π’€π’Šπ’”π’•|𝒔 = π’π’π’π’”π’†π’„π’–π’“π’Šπ’•π’Šπ’›π’†π’“, 𝒕 = 𝒂𝒇𝒕𝒆𝒓 πŸπŸŽπŸŽπŸ–] = πœΈπ’π’π’π’”π’†π’„π’–π’“π’Šπ’•π’Šπ’›π’†π’“+ π€π’‘π’π’”π’•πŸŽπŸ– (2.65)

Therefore, the difference between the securitizers’ performance before

and after 2008 is:

𝑬[π’€π’Šπ’”π’•|𝒔 = π’π’π’π’”π’†π’„π’–π’“π’Šπ’•π’Šπ’›π’†π’“, 𝒕 = 𝒂𝒇𝒕𝒆𝒓 πŸπŸŽπŸŽπŸ–] βˆ’

𝑬[π’€π’Šπ’”π’•|𝒔 = π’π’π’π’”π’†π’„π’–π’“π’Šπ’•π’Šπ’›π’†π’“, 𝒕 = 𝒃𝒆𝒇𝒐𝒓𝒆 πŸπŸŽπŸŽπŸ–] = π€π’‘π’π’”π’•πŸŽπŸ–βˆ’ π€π’‘π’“π’†πŸŽπŸ– (2.66) Finally, the Difference-in-Difference strategy allows us to compare the change in the performance of securitizers with the change in the performance of non-securitizers. The population Difference-in-Difference is:

{𝑬[π’€π’Šπ’”π’•|𝒔 = π’”π’†π’„π’–π’“π’Šπ’•π’Šπ’›π’†π’“, 𝒕 = 𝒂𝒇𝒕𝒆𝒓 πŸπŸŽπŸŽπŸ–] βˆ’ 𝑬[π’€π’Šπ’”π’•|𝒔 = π’”π’†π’„π’–π’“π’Šπ’•π’Šπ’›π’†π’“, 𝒕 = 𝒃𝒆𝒇𝒐𝒓𝒆 πŸπŸŽπŸŽπŸ–]} βˆ’

{𝑬[π’€π’Šπ’”π’•|𝒔 = π’π’π’π’”π’†π’„π’–π’“π’Šπ’•π’Šπ’›π’†π’“, 𝒕 = 𝒂𝒇𝒕𝒆𝒓 πŸπŸŽπŸŽπŸ–] βˆ’

𝑬[π’€π’Šπ’”π’•|𝒔 = π’π’π’π’”π’†π’„π’–π’“π’Šπ’•π’Šπ’›π’†π’“, 𝒕 = 𝒃𝒆𝒇𝒐𝒓𝒆 πŸπŸŽπŸŽπŸ–]} = 𝜹 (2.67)

The advantages of the Difference-in-Difference method are stated as follows. First, it is easy to calculate standard errors under this framework. Second, it allows researchers to control for other variables which may reduce the residual variance, which could also lead to smaller standard errors. Third, it is also easy to include multiple periods. Last, researchers can study treatments with different treatment intensity.

A typical regression model that can be estimated under the Difference-in- Difference framework is presented as follows:

π‘Άπ’–π’•π’„π’π’Žπ’†π’Šπ’•= 𝜷𝟎+ πœ·πŸπ‘»π’“π’†π’‚π’•π’Žπ’†π’π’•π’Š+ πœ·πŸπ‘·π’π’”π’•π‘Ίπ’‰π’π’„π’Œπ’•+ πœ·πŸ‘(π‘»π’“π’†π’‚π’•π’Žπ’†π’π’• Γ— π‘·π’π’”π’•π‘Ίπ’‰π’π’„π’Œ)π’Šπ’•+ 𝜺 (2.68)

Where π‘»π’“π’†π’‚π’•π’Žπ’†π’π’• is the dummy if a bank confirmed as a securitizer, while

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