Forecasting an Accumulated Series Based on Partial Accumulation II: A New Bayesian Method for Short Series With Stable Seasonal Patterns

Manuel Mendoza, Department of Statistics

Enrique de Alba, Department of Actuarial Science and Insurance

Instituto Tecnológico Autónomo de México (ITAM) Río Hondo No. 1 México, D.F. 01000

June 19, 2003

Abstract

In this paper, the problem of forecasting a time series with only a small amount of data is addressed within the Bayesian framework. The quantity to be predicted is the accumulated value of a positive and continuous variable for which some partially accumulated data has been observed. These conditions appear in a natural way in the prediction of sales of style goods and many other situations. A very simple model is proposed to describe the relation between the partial and the total values of the variable to be forecasted under the assumption of stable seasonality. Stable seasonality is specified in stochastic terms. Analytic results are obtained for both, the point forecast and the entire posterior predictive distribution. The proposed technique does not involve any approximation. Moreover, it allows the use of non-informative priors so that implementation may be automatic. The procedure works well when standard methods cannot be applied due to the reduced number of observations. It also improves on previous results published by the authors. Some real examples are included.

1 Introduction

Let us assume that for any given time t > t_{0}, Z(t) represents the accumulated value of the variable Z over the interval (t_{0}, t], where t_{0} is an arbitrary origin. There are many examples of phenomena which are recorded according to this accumulation mechanism.

Usually, observation takes place in a collection of equally spaced time points. Moreover, it often happens that only the most recent accumulation is recorded at each time point. In other words, the series of interest {X_{i}} is such that

X_{i} = Z(t_{0} + Δ × i) − Z(t_{0} + Δ × (i − 1)); i = 1, 2, ..., n

where Δ > 0 is the common length of the observation intervals. The typical problem here is that of forecasting X_{n+1} on the basis of the past observations X_{1}, X_{2}, ..., X_{n}. However, given the nature of the accumulation process, it is sometimes possible to obtain additional information in the form of a partially accumulated series {Y_{i}^{α}}. The structure of a series of this kind is the following

Y_{i}^{α} = Z(t_{0} + α × Δ × i) − Z(t_{0} + Δ × (i − 1)); i = 1, 2, ..., n

where α ∈ (0, 1) is a fixed, known value and the convention Z(t_{0}) = 0 is adopted. Thus, for each one of the observed intervals, of length Δ, the accumulation over the initial part, of length α × Δ is also recorded. For example, in some applications it might happen that X_{i} refers to the accumulated value of Z over the i − th year whereas Y_{i}^{α} stands for the accumulated value of Z over the first month (α = 1/12), the first quarter (α = 1/4) or the first semester (α = 1/2) of the year. In any case, for each fixed value of α, we have a partially accumulated series.

Most of the forecasting techniques only use X_{1}, X_{2}, ..., X_{n} to forecast X_{n+1} despite the fact that the series Y_{1}^{α}, Y_{2}^{α}, .., Y_{n}^{α} and Y_{n+1}^{α} might contain very valuable information. In particular, it is obvious that if α is close to 1, Y_{n+1}^{α} must be close to X_{n+1} but, even for other values of α, some empirically verifiable structure might be considered to relate Y_{i}^{α} and X_{i} (i = 1, 2, ..., n+1) in a useful way.

In any case, the use of partially accumulated series has been explored in a number of papers (Oliver 1987, de Alba 1988, Lenk 1992, Guerrero and Elizondo 1997, and Chen and Fomby 1999, among others) and has proved to be particularly useful when the series exhibit a stable seasonal pattern and only a few past observations are available (de Alba and Mendoza 1996, 2001).

A stable seasonal pattern is one of those empirically verifiable structures that we have already mentioned. The idea is that, for a fixed value of α, Y_{i}^{α} represents a proportion of X_{i} which, in a well defined stochastic sense, is supposed to be stable for i = 1, 2, ..., n, n + 1. We will propose an specific definition for this concept in the following section.

A first generation of Bayesian analyses for this problem was produced by Oliver (1987), de Alba (1988) and Lenk (1992) where some rather involved models and several approximations were proposed. More recently, de Alba and Mendoza (1996), (2001) have shown that the use of the partially accumulated series is crucial when only a few past observations are available (n = 2, for example) and on this basis they have argued that simpler models where no approximations are required must be used. In particular, de Alba and Mendoza (1996) deals with the case where Z is a discrete, nonnegative random variable and reduces the problem of forecasting X_{n+1} with both sets of observations X_{1}, X_{2}, ..., X_{n} and Y_{1}^{α}, Y_{2}^{α}, .., Y_{n+1}^{α} to that of making inferences on the parameter N of a binomial distribution bin(θ_{α}, N). On the other hand, de Alba and Mendoza (2001) explore the case where Z is a positive continuous random variable and a posterior predictive distribution for X_{n+1} is obtained, using an inverted Pareto distribution which leads to highly competitive pointwise forecasts despite the fact that the predictive model has no moments due to its very heavy right tail. A rather odd characteristic of that model is that the corresponding highest posterior density predictive intervals are, for most practical purposes, useless.

Here, the case of Z positive and continuous is revisited and a more flexible model which maintains the level of simplicity and does not require any kind of approximation is proposed. By using some real data sets it is shown that pointwise forecasts are comparable to those obtained in de Alba and Mendoza 2001, but the posterior predictive intervals are very much improved.

2 The Model

Let us elaborate on the problem of forecasting X_{n+1} and on the structure to be used to incorporate the partially accumulated data. On the basis of the positive, continuous series {X_{1}, X_{2}, ..., X_{n}}, {Y_{1}^{α}, Y_{2}^{α}, ..., Y_{n}^{α}} and the observation Y_{n+1}^{α} we must forecast the value of X_{n+1}. Specifically, we are interested in the conditional distribution of X_{n+1} given Y_{n+1}^{α} = y. Initially, we can assume that this distribution belongs to a particular family of probability models F parametrized by a vector \underline{θ} and thus, if \underline{θ} were known, we could produce the required inferences on the basis of the completely specified distribution P (X_{n+1}|Y_{n+1}^{α} = y, \underline{θ}).

In order to obtain an appropriate solution we must propose a family of distributions and then an inference procedure for the case where, as usual, the parameter \underline{θ} is unknown. Let us deal with the first issue by noticing that since the underlying variable Z is positive we know that X_{n+1} ≥ Y_{n+1}^{α}. Even more, we can write

X_{n+1} = Y_{n+1}^{α} + (X_{n+1} − Y_{n+1}^{α})

= Y_{n+1}^{α} \{1 + \frac{(X_{n+1} − Y_{n+1}^{α})}{Y_{n+1}^{α}}\} ,

so that given Y_{n+1}^{α} = y (y > 0), we have

X_{n+1} = y \{1 + \frac{(X_{n+1} − y)}{y}\}

In other words, given Y_{n+1}^{α} = y, X_{n+1} can be obtained as a linear transformation of the ratio

W_{n+1} = \frac{(X_{n+1} − y)}{y}

which describes the relative magnitude of the (yet) unobserved accumulation as compared with the (already) observed part. As a consequence, from the expression

X_{n+1} = y(1 + W_{n+1}),

it is clear that, given Y_{n+1}^{α} = y, all we need is a distribution for W_{n+1}. In order to do so let us extend the argument to the rest of the available observations. In general, given Y_{i}^{α} = y_{i}; i = 1, ..., n + 1, we have

X_{i} = y_{i} \{1 + \frac{(X_{i} − y_{i})}{y_{i}}\}

= y_{i}(1 + W_{i}); i = 1, 2, ..., n + 1,

where W_{i} describes the ratio of the unobserved part of X_{i}, at the point of time t = t_{0} + α × Δ × i, as compared with the already observed part.

Here, we will introduce a precise formulation of the idea of a stable seasonal pattern. In fact, we shall say that the accumulation mechanism follows a stable seasonal pattern if the random variables W_{1}, W_{2}, ..., W_{n}, W_{n+1} are independent and identically distributed according to a probability distribution P (W |\underline{ϕ}).

The basic idea behind this definition is that the behavior of the unobserved/observed ratios is the same, in a stochastic sense, over the time intervals to be considered.

More technically, the i.i.d. assumption allows us to use the observed ratios w_{1}, w_{2}, ..., w_{n}, in a very simple way to obtain information regarding the parameter \underline{ϕ} and thus, producing forecasts for the value of W_{n+1}. Finally, the relation of W_{n+1} with X_{n+1} can be used to get the required forecast for X_{n+1} (given Y_{n+1}^{α} = y).

Now, taking into account the nature of the variable W_{n+1} (continuous with support W = (0, ∞)) and in order to maintain the idea of a simple model, we shall assume that W_{1}, W_{2}, .., W_{n}, W_{n+1} are i.i.d. random variables following a log-Normal distribution with parameter \underline{ϕ} = (μ, σ).

Equivalently, if we define

U_{i} = log(W_{i}) i = 1, 2, ..., n, n + 1

we shall assume that U_{1}, U_{2}, ..., U_{n}, U_{n+1} are i.i.d. random variables following a Normal (μ, σ) distribution.

3 Prior-Posterior Analysis

Under the Normality assumption for U, application of the Bayesian paradigm is absolutely straightforward in order to obtain the posterior predictive distribution for U_{n+1} given the past data u_{1}, ..., u_{n}. As for the prior distribution and recalling that this proposal is expected to be particularly useful when only a small amount of past data is available, we propose to use a jointly noninformative prior for μ and τ = 1/σ^{2} (Bernardo and Smith 1994, page 440)

π(μ, τ) ∝ τ^{−1}.

If we denote \underline{u}_{(n)} = (u_{1}, u_{2}, ..., u_{n}), then we know that

p(μ, τ|\underline{u}_{(n)}) = p(μ|τ, \underline{u}_{(n)})p(τ|\underline{u}_{(n)})

= N(μ|\overline{u}, nτ) Gamma (τ|\frac{n − 1}{2}, \frac{nS_{uu}}{2} )

where \overline{u} = \frac{1}{n} ∑_{i=1}^{n} u_{i} and nS_{uu} = ∑_{i=1}^{n}(u_{i} − \overline{u})^{2}. As for the predictive distribution

of U_{n+1} we have that

p (u_{n+1}|\underline{u}_{(n)}) = ∫ p (u_{n+1}|μ, τ) p (μ, τ|\underline{u}_{(n)}) dμdτ

and then

p (u_{n+1}|\underline{u}_{(n)}) = Student (u_{n+1}|\overline{u}, (\frac{n − 1}{n + 1}) σ̂^{−2}, n − 1) ,

where σ̂^{2} = S_{uu}. Now recalling that U_{i} =log(W_{i}) ; i = 1, 2, ..., n + 1 we have that the predictive distribution for W_{n+1} is immediately obtained as

p (w_{n+1}|\underline{w}_{(n)}) = log-Student (w_{n+1}|\overline{u}, (\frac{n − 1}{n + 1}) σ̂^{−2}, n − 1) .

Finally, using the fact that given Y_{n+1}^{α} = y,

X_{n+1} = y(1 + W_{n+1})

we have that

P (X_{n+1} ≤ c|y, \underline{w}_{(n)}) = P (\frac{X_{n+1} − y}{y} ≤ \frac{c − y}{y} |y, \underline{w}_{(n)} )

= P (W_{n+1} ≤ \frac{c − y}{y} |y, \underline{w}_{(n)} ) ,

or equivalently, given Y_{n+1}^{α} = y, X_{n+1} follows a nonstandard log-Student distribution with n − 1 degrees of freedom. Incidentally, it is interesting to notice that this predictive distribution for X_{n+1}, given Y_{n+1}^{α} = y has no moments, a characteristic in common with the predictive distribution obtained in de Alba and Mendoza (2001). In any case, the corresponding q-quantile for X_{n+1}, given Y_{n+1}^{α} = y, is given by

X_{n+1}^{(q)} = y + y [ ∏_{i=1}^{n} w_{i}]^{1/n} [exp (t_{(n−1)}^{(q)})]^{(1+ \frac{1}{n})^{1/2}σ̃^{u}} ,  (1)

where t_{(n−1)}^{(q)} is the corresponding quantile of a standard Student distribution with n − 1 degrees of freedom and σ̃_{u}^{2} = \frac{1}{n−1}Σ(u_{i} − \overline{u})^{2}. Clearly, these quantiles can be used as forecasts for X_{n+1} and, in addition, we can compute predictive probability intervals. The specific value of q to be used will depend on the particular loss function involved although in the following section we will elaborate on an empirical approach to select this value.

4 Implementation

In this section we establish some criteria which are necessary to make our method operational. On the one hand we need to have a practical criterion to decide how stable the seasonal pattern is for a given set of data. One way to do it is of course to look at the autocorrelation function, or at the partial autocorrelation function, or at the periodogram, or at the spectrum, etc. of the series. However, as mentioned in de Alba and Mendoza (2001), with only a few observations in a series of, say, monthly data they will not be of much help.

We have found that in practical situations it is often possible to observe Y_{i}^{α} for several values of α and thus, the cumulative proportions P_{i}^{α} = Y_{i}^{α}/X_{i} can be plotted as a (non decreasing) function of α for a fixed i. In de Alba and Mendoza (2001) the series were assumed to have a stable seasonal pattern if P_{1}^{α}, ..., P_{n}^{α} were i.i.d.. Thus, the cumulative proportion curves were plotted for i = 1, ..., n and on that basis the validity of the assumption was assessed. If the lines lie very close together, then we have empirical evidence in favor of a stable pattern. On the other hand, if they wander in separate directions then this is an indication of non-stability. In this paper the stable seasonal pattern is defined in terms of W_{i} instead of P_{i}^{α}. However, it follows immediately that these two variables are closely related, since

W_{i} = 1/P_{i}^{α} − 1.

As a consequence, both characterizations of stable seasonality are equivalent and thus, the same cumultative curves as in de Alba and Mendoza (2001) can be plotted in our case to provide empirical insight on the assumption. Alternatively, we can plot W_{i}^{α} = (X_{i} − Y_{i}^{α})/Y_{i}^{α} as a function of α for i = 1, ..., n and, again stable seasonality will be considered to be a reasonable assumption if the curves are close to one another. This will be the procedure used in this paper.

5 Applications

In this section we apply the results of Section 2 to time series that are known (conceptually) to be seasonal: Chequing Accounts in the Mexican Banking System (1968-1971), Real Monthly Administration Expenses of the Mexican Banking System (1992-1994), Monthly Average Residential Electricity Usage in Iowa City(1976-1978), Monthly Car Sales in Quebec (1966-1968) and Inflation in Mexico(1994-1996). The first one is taken from de Alba and Nieto de Pascual (2002), the second one is from Guerrero and Elizondo (1997), the third and fourth are from Abraham and Ledolter (1983), and the last one is from Banco de México (1997). For the sake of brevity each one is given a short name: Cheques, Expenses, Electricity, Cars and Inflation, respectively. Although there is more information available for each one of the series, we only use the last three years available for every one. In addition, we pretend that we do not have the data for the very last year and use the information on the first two to forecast the total as indicated in the previous sections. Moreover, the accumulation pattern is recorded for each month and thus, eleven forecasts can be produced for the third year (α = r/12; r = 1, ..., 11). Figure 1 shows the plot of the three years of data for each series. Notice that the first two sets of data have a trend in addition to the seasonality; in the last one the seasonality is not evident. The other two seem to present stable seasonality. Figure 2 shows the three years ’overlaid’ for each series. Notice there are two observations missing in 1994 for ’Expenses’, but this was the available data and has been taken into account in the analysis. These graphs reinforce what we observe in the first one. As stated, we only use data from two years and thus, for every fixed value of α (or equivalently, r), we have two observed values of the random variable W. However, we do have w_{1} and w_{2} for each month of the year. In Figure 3, according to the final paragraph of Section 4, we plot w_{ir} and w_{2r}, r = 1, ..., 11 for each one of the series. The only one that shows a clear deviation from the assumption that W_{i}’s are i.i.d., as required by our method, is Inflation in Mexico. In the other four it is essentially impossible to distinguish the two lines, except for ”Expenses”, that shows a very slight discrepancy between w_{1r} and w_{2r} in the first month, i.e. for r = 1.

In each one of the examples we will compare the performance of our procedure with that of de Alba and Mendoza (2001). For this purpose we shall take advantage of two facts: first, we know the true value of X_{n+1} and second, in each case we can produce eleven forecasts, depending on the number of months used to calculate the partial accumulation (the only exception is the case of ”Expenses”, where only ten forecasts were produced). So we have, for each procedure to be compared, a sequence of forecasts (X_{n+1,r}^{(q)} ; r = 1, 2, ..., 11) for the value X_{n+1}, where q is assumed to be the same for different months. Under such circumstances, we propose the following measure to evaluate the forecasting procedures

MSE(q) = ∑_{r=1}^{11}(X_{n+1,r}^{(q)} − X_{n+1})^{2}/11.  (2)

Since the forecast X_{n+1,r}^{(q)} can be updated to X_{n+1,r+1}^{(q)} once the user gets the information for the (r + 1) − th month, we think that MSE(q) produces an overall assessment of each procedure’s performance.

The value of q in Equation (2) must be specified by the statistician. As previously mentioned, it can be indirectly determined by choosing a specific loss function. In any case, the choice of q need not be the same in all problems. In this section, we analyze the behavior of the forecasts for alternative values of q, using the same q at different partial accumulations for each series.

Table 1 shows the values of the minimum MSE(q) and the corresponding value of q. For reference, we have included the results for all the series. Figure 4, shows the graph of the MSE(q) vs q, for all the series. Notice that for three of the series the optimal values of q are very similar and close to q = 0.7. For the series on ”Cheques” the value is larger, 0.825, and for ”Inflation” it is lower, 0.4. The small value in the case on ”Inflation” points in the direction that when the ’stable seasonality’ is not evident, as was clear in this case from the graph in Figure 3, then probably your best point estimate will be close to the median. In de Alba and Mendoza (2001) only two of these data sets were analyzed. In both cases the the MSE(q) attained its minimum at q = 0.557. The corresponding values are 5442216 (Electricity) and 6123 (Expenses). In order to complete Table 1, we applied the procedure proposed there to the other series. The optimal value of q was also 0.557 for both ”Cars” and ”Cheques”, and 0.525 for ”Inflation”.

We now look in detail at the point forecasts and probability intervals for the series ”Electricity” and ”Expenses”, since these series were analyzed in de Alba and Mendoza (2001). Table 2 shows the point estimates for ”Electricity” obtained with q = 0.7, using the new method, as well as the upper and lower limits corresponding to a 0.95 probability interval. Table 2 also includes results from the old method. The new method intervals are displayed in Figure 5, panel a), along with the pointwise forecasts for this series. The thick horizontal line is the observed value for X_{n+1}. Table 3 parallels Table 2, but with the results for ”Expenses”, which are plotted in Figure 6 for the new method. In general terms, we can conclude that the pointwise forecasts are similar for both methods. However, the corresponding intervals are dramatically different. In fact, it can be verified that the intervals for the old method are, many times larger, compared to those of the method proposed in this paper. In addition, in practically all cases, the 0.95 probability intervals for the new method include the true value of X_{n+1} for both series.

6 Final Remarks

The general problem of producing forecasts, both pointwise and intervals, for a time series where only a very small number of past observations are available, entails a formidable challenge. In such a situation, specific knowledge regarding the process underlying the series might be extremely useful. In the case of a positive, accumulated series, information on the accumulation process turns out to be crucial. In this paper we have introduced a very simple structure based on a log-Normal distribution which not only reproduces, for the examples considered, the pointwise forecasts previously obtained for these series but also dramatically improves the corresponding interval forecasts.

The analysis is entirely Bayesian, only noninformative priors are used in order to obtain conclusions based on the available data, and every result is derived explicitly without any approximation.

In a sense, the proposed method might be considered as a second generation technique as compared to that in de Alba and Mendoza (2001). There, the authors argued extensively in favor of a noninformative Bayesian analysis of simple models. The idea of maximizing the final contribution of the limited set of data as well as that of avoiding and kind of approximation was stressed. Here, we have gone one step beyond. Substitution of the one-parameter model used in de Alba and Mendoza (2001) by the two-parameters log-Normal distribution in this paper has provided the modeling process with enough flexibility to obtain useful intervals while the analysis is basically maintained on the same level of complexity. It is worth noting that the useless intervals obtained in de Alba and Mendoza (2001) were attributed to the fact that the corresponding posterior predictive distribution was a heavy tailed model where no moment existed. Now, we know that with a more flexible structure we can also get heavy tailed predictive distributions. In fact, our new procedure leads to log-Student distributions and, for the examples with n = 2, to log-Cauchy distributions. Hence, we have predictive models whose moments do not exist and nevertheless, are able to produce useful intervals.

In this paper, the log-Normal likelihood was introduced, as in many other cases, as a model to work with transformed instead of the original set of data. Obviously, some other transformation, apart of the logarithmic, could be assayed. This idea is under current investigation and will be reported elsewhere.

References

[1] Abraham, B., and Ledolter, J. (1983), Statistical Methods for Forecasting, New York: Wiley.

[2] Banco de México, S.A. (1997), Indicadores Económicos, enero-febrero, Mexico City: Banxico. Also http://www.banxico.org.mx

[3] Bernardo, J.M. and Smith, A.F.M. (1994), Bayesian Theory, New York: Wiley.

[4] Box, G. E. P., and Jenkins, G. M. (1970), Time Series Analysis, Forecasting and Control of Discrete Time Series, San Francisco: Holden-Day.

[5] Box, G. E. P., and Tiao, G. C. (1973), Bayesian Inference in Statistical Analysis, Reading, MA: Addison-Wesley.

[6] Chen, R., and Fomby, B. T. (1999), ”Forecasting With Stable Seasonal Pattern Models With Application to Hawaiian Tourism Data”, Journal of Business and Economic Statistics, 17, 497-504.

[7] de Alba, E. (1988), ”Pronóstico Bayesiano de Agregados en Procesos Estacionales Estables”, Revista Estadística, 2, 1-6.

[8] de Alba, E., and Mendoza, M. (1996), ”A Discrete Model for Bayesian Forecasting With Stable Seadonal Patterns”, in Advances in Econometrics Vol. 11: Bayesian Methods Applied to Time Series Data, eds. T. B. Fomby and R. C. Hill, Stamford, CT: JAI Press, pp. 267-281.

[9] de Alba, E. and Mendoza, M. (2001), ”Forecasting an Accumulated Series Based on Partial Accumulation: A Bayesian Method for Short Series With Seasonal Patterns”, Journal of Business and Economics Statistics Vol. 19, No.1, 95-106.

[10] de Alba, E. and Nieto de Pascual, J. (2002), ”Forecasting in Processes With Stable Seasonal Patterns”, Revista Agrociencia 29,6, forthcoming

[11] Guerrero, V. M., and Elizondo, J. A. (1997), ”Forecasting a Cumulative Variable Using its Partially Accumulated Data”, Management Science, 43, 879-889.

[12] Lenk, P. J. (1992), ”Hierarchical Bayes Forecasts of Multinomial Dirichlet Data Applied to Coupon Redemptions”, Journal of Forecasting, 603-619.

[13] Minitab, Inc. (1994), MINITAB, State College, PA: Author.

[14] Mood, A. M., Graybill, F. A., and Boes, D. C. (1974), Introduction to the Theory of Statistics (3rd ed.), New York. McGraw-Hill.

[15] Oliver, R. M. (1987), ”Bayesian Forecasting With Stable Seasonal Patterns”, Journal of Business and Economic Statistics, 5, 77-86.

Figure 1: Data sets for the examples. 11 Figure 2: Overlaid years.

12 Figure 3: Plots of W_{1} and W_{2}.

13 Figure 4: MSE(q) for each example.

14 Figure 5: Pointwise forecasts and intervals.

15

Table 1. Optimal values of q.

Series Optimal q MSE(q) MSE(2001) Ratio
Cheques  0.825  2086814  1237790  1.686
Expenses  0.700  5110450  5442216  0.939
Electricity  0.700  2419  6123  0.395
Cars  0.625  21.2  20.2  1.052
Inflation  0.425  12.3  2.0  6.028

Table 2. Electricity use in Iowa (Kw/h): Comparison of Results.

New Bayesian Method Old Bayesian Method
q  q
month 0.025 0.7 0.975  0.025 0.557 0.975
1  6063 6082 6098  569 6081 9.E+13
2  5933 6030 6118  1086 6025 1.E+11
3  5678 6147 6610  1557 6120 5.E+09
4  5190 6151 7243  1973 6092 4.E+08
5  5304 6105 6997  2356 6056 7.E+07
6  5121 6058 7210  2826 6001 1.E+07
7  4532 6139 9494  3496 6009 1.E+06
8  4716 6088 9630  4109 5968 2.E+05
9  5604 6155 6925  4752 6116 74414
10  5752 6210 6932  5218 6179 32928
11  6132 6145 6163  5630 6147 14674

Observed value: 6118 Table 3. Expenses of the Mexican Bank System (Thousands of New Pesos):

Comparison of Results.

New Bayesian Method  Old Bayesian Method
q  q
Month 0.025  0.7  0.975  0.025  0.557  0.975
1  67171  110070  173982  8534  108474 8.E+15
2  83690  109237  140075 16984 108638  9.E+12
3  103864 108254  112375 26048 108784  1.E+11
4  104357 107261  109950 34460 107711  8.E+09
5  100849 106962  112946 42888 107104  9.E+08
6  100073 105870  111607 51245 105923  1.E+08
7  100567 105236  109838 59764 105268  3.E+07
8  93231  105959  122621 68676 105359  7.E+06
9  94200  104629  118781 76234 104104  2.E+06
10  98971  104533  111304 85179 104290 765390.86

Observed value: 108152