5.2 6 Testable Implications

6 Testable Implications

  1. Statistical significance of \(f(d)\)

  2. Is there a better estimate for \(q\) than \(f \times c\)

  3. Check residuals against \(c(w,d)\)

    • Test for Mack assumption 1 (4.1)?

  4. Stability of \(f(d)\) down the column

    • Test for Mack assumption 1 (4.1)?

  5. No correlation among columns

    • Test for Mack assumption 1 (4.1)

  6. No particularly high or low diagonals

    • Test for Mack assumption 2 (4.2)

5.2.1 Goodness of Fit Measurement

Compare different fit of the models based on adjusted \(SSE\) (actual vs projected excluding 1st column)

Adjusted SSE

\[\begin{equation} \dfrac{SSE}{(n-p)^2} \tag{5.1} \end{equation}\]

Akaike Information Criterion

\[\begin{equation} AIC \approx SSE \times e^{2p/n} \tag{5.2} \end{equation}\]

Bayesian Information Criterion

\[\begin{equation} BIC \approx SSE \times n^{p/n} \tag{5.3} \end{equation}\]

Remark.

  • \(n =\) # of predicted data points EXCLUDING 1st column

    • Exclude because when we do reserving we don’t predict anything from the first column

    • So usually equals number of cells in the triangle excluding first column

  • \(p =\) # of parameters

  • \(SSE = \sum (A - E)^2\)

    • Here you exclude the first column when calculating the difference
  • Venter use the adjusted SSE as the AIC can be too permissive of over parameterization for large data sets

SSE calculation can be done with the table features on TI-30XS

  • Plug in each the actual and projected triangle into L1 and L2 (make sure the cells match)

  • Calculate L3 = (L1 - L2)2

  • Calculate ∑L3