4.2 Analyzing the Time-Stability of Transition Probabilities


4.2.1 Aggregation over Periods

We assume that migration data are given for $ m$ periods. This data consist in $ m$ matrices of migration counts $ \mathbf{C}(t)$ for $ t= 1,\ldots,m$ each of type $ (d-1) \times d$. The generic element $ c_{jk}(t)$ of the matrix $ \mathbf{C}(t)$ is the number of migrations from $ j$ to $ k$ in period $ t$. These matrices may be computed from $ m$ data sets of migration events.

An obvious question in this context is whether the transition probabilities can be assumed to be constant in time or not. A first approach to analyze the time-stability of transition probabilities is to compare the estimated transition probabilities per period for $ m$ periods with estimates from pooled data.

The aggregated migration counts from $ m$ periods are

$\displaystyle c_{jk}^{+} \stackrel{\mathrm{def}}{=}\sum_{t=1}^m c_{jk}(t)$ (4.10)

which are combined in the matrix

$\displaystyle \mathbf C^{+} \stackrel{\mathrm{def}}{=}\sum_{t=1}^m \mathbf C(t) $

of type $ (d-1) \times d$. The migration rates computed per period

$\displaystyle \hat p_{jk}(t) \stackrel{\mathrm{def}}{=}\frac{c_{jk}(t)}{n_j(t)}, \quad t = 1,\ldots, m$ (4.11)

with

$\displaystyle n_j(t) \stackrel{\mathrm{def}}{=}\sum_{k=1}^{d} c_{jk}(t) $

have to be compared with the migration rates from the pooled data. Based on the aggregated migration counts the estimated transition probabilities

$\displaystyle \hat p_{jk}^{+} \stackrel{\mathrm{def}}{=}\frac{c_{jk}^{+}}{{n_j}^{+}}\,$ (4.12)

with

$\displaystyle n_j^{+} \stackrel{\mathrm{def}}{=}\sum_{k=1}^{d} c_{jk}^{+} = \sum_{t=1}^m n_j(t) ,
\quad j = 1, \ldots, d-1 $

can be computed.


4.2.2 Are the Transition Probabilities Stationary?

Under the assumption of independence for the migration events the vector of migration counts $ (c_{j1}(t),\ldots
c_{jd}(t))$ starting from $ j$ is in each period $ t$ a realization from a multinomial distributed random vector

$\displaystyle (\tilde c_{j1}(t),\ldots, \tilde c_{jd}(t)) \sim {\rm Mult}(n_j(t);p_{j1}(t),\ldots, p_{jd}(t)), $

where $ p_{jk}(t)$ denotes the transition probability from $ j$ to $ k$ in period $ t$. For fixed $ j \in \{1,\ldots,d-1 \} $ the hypothesis of homogeneity

$\displaystyle H_0: p_{j1}(1) = \ldots = p_{j1}(m), p_{j2}(1) = \ldots =
p_{j2}(m), \ldots, p_{jd}(1) = \ldots = p_{jd}(m)
$

may be tested with the statistic

$\displaystyle X_j^2 = \sum_{k=1}^d \sum_{t=1}^m \frac{\left[ \tilde c_{jk}(t) - n_j(t) \hat p_{jk}^{+}\right]^2}{ n_j(t) \hat p_{jk}^{+} }.$ (4.13)

This statistic is asymptotically $ \chi^2$-distributed with $ (d-1)(m-1)$ degrees of freedom under $ H_0$. $ H_0$ is rejected with approximative level $ \alpha$ if the statistic computed from the data is greater than the $ (1-\alpha)$-quantile of the $ \chi^2$-distribution with $ (d-1)(m-1)$ degrees of freedom.

The combined hypothesis of homogeneity

$\displaystyle H_0: p_{jk}(t) = p_{jk}(m), \quad t = 1,\ldots, m-1, \quad j =
1,\ldots, d-1, \quad k = 1,\ldots,d
$

means that the matrix of transition probabilities is constant over time. Therefore, the combined null hypothesis may equivalently be formulated as

$\displaystyle H_0: \mathbf{P}(1) = \mathbf{P}(2) = \ldots = \mathbf{P}(m),$

where $ \mathbf{P}(t)$ denotes the transition matrix at $ t$ with generic element $ p_{jk}(t)$. This hypothesis may be tested using the statistic

$\displaystyle X^2 = \sum_{j=1}^{d-1} X_j^2,$ (4.14)

which is under $ H_0$ asymptotically $ \chi^2$-distributed with $ (d-1)^2(m-1)$ degrees of freedom. The combined null hypothesis is rejected with approximative level $ \alpha$ if the computed statistic is greater than the $ (1-\alpha)$-quantile of the $ \chi^2$-distribution with $ (d-1)^2(m-1)$ degrees of freedom (Bishop et al.; 1975, p. 265).

This approach creates two problems. Firstly, the two tests are based on the assumption of independence. Secondly, the test statistics are only asymptotically $ \chi^2$-distributed. This means that sufficiently large sample sizes are required. A rule of thumb given in the literature is $ n_j(t) \hat p_{jk}^{+} \geq
5$ for all $ j$ and $ k$ which is hardly fulfilled in the context of credit migrations.

The two $ \chi^2$-statistics in (4.13) and (4.14) are of the Pearson type. Two other frequently used and asymptotically equivalent statistics are the corresponding $ \chi^2$-statistics of the Neyman type

$\displaystyle Y^2_j =
\sum_{k=1}^d \sum_{t=1}^m
\frac{ \left[\tilde c_{jk}(t) -...
...t p_{jk}^{+}\right]^2
}{ \tilde c_{jk}(t) },\quad Y^2 = \sum_{j=1}^{d-1} Y^2_j
$

and the $ \chi^2$-statistics

$\displaystyle G_j^2 = 2 \sum_{k=1}^d \sum_{t=1}^m \tilde c_{jk}(t)
\ln \left[ \...
...jk}(t) }{ n_j (t)\hat p_{jk}^{+} } \right],\quad
G^2 =
\sum_{j=1}^{d-1} G_j^2,
$

which results from Wilks log-likelihood ratio.

Considering the strong assumptions on which these test procedures are based on, one may prefer a simpler approach complementing the point estimates $ \hat p_{jk}(t)$ by estimated standard errors

$\displaystyle \hat\sigma_{jk}(t) = \sqrt{ \frac{\hat p_{jk}(t)(1-\hat
p_{jk}(t)) }{n_j(t)}}
$

for each period $ t \in\{1,\ldots,m\}$. For correlated migrations the estimated standard deviation is computed analogously to (4.9). This may graphically be visualized by showing

$\displaystyle \hat p_{jk}^{+}, \quad \hat p_{jk}(t), \quad \hat p_{jk}(t) \pm 2\hat\sigma_{jk}(t), \quad t = 1,\ldots, m$ (4.15)

simultaneously for $ j = 1, \ldots, d-1$ and $ k= 1,\ldots, d$.


4.2.3 Computation and Quantlets

The quantlet 10148 XFGRatMig3.xpl computes aggregated migration counts, estimated transition probabilities and $ \chi^2$-statistics. The call is out = XFGRatMig3(c, rho, s), where c is a $ (d-1) \times d \times m$ array of counts for $ m$ periods and rho is a non-negative correlation parameter. For $ \texttt{rho} = 0$ the independent case is computed, compare Section 4.1.4. The last input parameter s controls the accuracy of the computation, see Section 4.1.4.

The result is assigned to the variable out, which is a list containing:

The matrices out.cagg, out.etpagg and out.etp have components given by (4.10), (4.12) and (4.11). The elements of out.esdagg and out.esd result by replacing $ \hat p_{jk}$ in (4.9) by $ \hat p^+_{jk}$ or $ \hat p_{jk}(t)$, respectively. The matrix out.chi contains in the first row the statistics from (4.13) for $ j = 1, \ldots, d-1$ and (4.14). The second and third row gives the corresponding degrees of freedom and p-values.

The quantlet 10151 XFGRatMig4.xpl (XFGRatMig4(etp, esd, etpagg)) graphs migration rates per period with estimated standard deviations and migration rates from pooled data. The inputs are:

The output consists of $ (d-1)d$ graphics for $ j = 1, \ldots, d-1$ and $ k= 1,\ldots, d$. Each graphic shows $ t= 1,\ldots,m$ at the $ x$-axis versus the four variables from (4.15) at the $ y$-axis.


4.2.4 Examples with Graphical Presentation

The following examples are based on transition matrices given by Nickell et al. (2000, pp. 208, 213). The data set covers long-term bonds rated by Moody's in the period 1970-1997. Instead of the original matrices of type 8 $ \times$ 9 we use condensed matrices of type 3 $ \times$ 4 by combining the original data in the $ d =4$ basic rating categories A, B, C, and D, where D stands for the category of defaulted credits.

The aggregated data for the full period from 1970 to 1997 are

\begin{displaymath}
\mathbf{C} = \left[
\begin{array}{rrrr}
21726 & 790 & 0 & 0 ...
....006 & 0.019 \\
0 & 0.102 & 0.709 & 0.189
\end{array}\right],
\end{displaymath}

where $ \mathbf{C}$ is the matrix of migration counts and $ \mathbf{\hat P}$ is the corresponding matrix of estimated transition probabilities. These matrices may be compared with corresponding matrices for three alternative states of the business cycles:

\begin{displaymath}
\mathbf{C}(1) = \left[
\begin{array}{rrrr}
7434 & 277 & 0 & ...
....008 & 0.024 \\
0 & 0.106 & 0.662 & 0.232
\end{array}\right],
\end{displaymath}

for the through of the business cycle,

\begin{displaymath}
\mathbf{C}(2) = \left[
\begin{array}{rrrr}
7125 & 305 & 0 & ...
....005 & 0.021 \\
0 & 0.115 & 0.702 & 0.183
\end{array}\right],
\end{displaymath}

for the normal phase of the business cycle, and

\begin{displaymath}
\mathbf{C}(3) = \left[
\begin{array}{rrrr}
7167 & 208 & 0 & ...
....005 & 0.011 \\
0 & 0.088 & 0.756 & 0.156
\end{array}\right],
\end{displaymath}

for the peak of the business cycle. The three categories depend on whether real GDP growth in the country was in the upper, middle or lower third of the growth rates recorded in the sample period (Nickell et al.; 2000, Sec. 2.4).

Figure: Example for 10292 XFGRatMig4.xpl
\includegraphics[width=1.42\defpicwidth]{RM01.ps}

In the following we use these matrices for illustrative purposes as if data from $ m=3$ periods are given. Figure 4.1 gives a graphical presentation for $ d =4$ rating categories and $ m=3$ periods.

In order to illustrate the testing procedures presented in Section 4.2.2 in the following the hypothesis is tested that the data from the three periods came from the same theoretical transition probabilities. Clearly, from the construction of the three periods we may expect, that the test rejects the null hypothesis. The three $ \chi^2$-statistics with $ 6 = 3(3-1)$ degrees of freedom for testing the equality of the rows of the transition matrices have $ p$-values 0.994, $ >0.9999$, and 0.303. Thus, the null hypothesis must be clearly rejected for the first two rows at any usual level of confidence while the test for the last row suffers from the limited sample size. Nevertheless, the $ \chi^2$-statistic for the simultaneous test of the equality of the transition matrices has $ 18 =
3^2\cdot(3-1)$ degrees of freedom and a $ p$-value $ >0.9999$. Consequently, the null hypothesis must be rejected at any usual level of confidence.

10296 XFGRatMig3.xpl

A second example is given by comparing the matrix $ \mathbf{\hat P}$ based on the whole data with the matrix $ \mathbf{\hat P}(2)$ based on the data of the normal phase of the business cycle. In this case a test possibly may not indicate that differences between $ \mathbf{P}$ and $ \mathbf{P}(2)$ are significant. Indeed, the $ \chi^2$-statistics for testing the equality of the rows of the transition matrices with 3 degrees of freedom have $ p$-values 0.85, 0.82, and 0.02. The statistic of the simultaneous test with 9 degrees of freedom has a $ p$-value of 0.69.