Elements of Forecasting - A Case Study: German Car Production

Table of Content

1.  Introduction page  3

2.  Introduction of the Series page  4

3.  Modelling a Univariate Model page  6

3.1  Trend Analysis page  6

3.2  Seasonal Analysis page  8

3.3  Cyclical Analysis page  9

3.4  Evaluation of the ARMA Model page  10

3.5  The Unit-Root Test page  11

3.6  The ARIMA Model page  11

4.  Modelling a Multivariate Model page  14

5.  Conclusion page  16

6.  Bibliography page  17

  Appendices A till G page  18

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1. Introduction

Forecasting is one of the mayor issue in today’s business world. Whether it concerns the

economic situation, stock prices, or production levels, a glance into the future would be very

valuable. By excluding uncertanties, expenses can be saved and revenues be generated.

Unnecessary or too little inventories, capacity standing idle or being short, missing raw

materials or too many employees are just some of the situations, which lead to lower profits.

Hence, perfect forecasts would be worth a lot of money. But, as the expression states, a

“perfect forecast” is a paradox, since the future will stay uncertain till the moment, where it

becomes the present. As the American philosopher Eric Hoffer once stated: “The only way to

predict the future is to have the power to shape the future”, which would take place in the

present. The one chance we have in making inferences about the future, is to incorporate

logic, intuition, and experience into models, which will then – if we are lucky, that is –

produce more or less accurate forecasts.

Forecasting, if pursued by professionals, relies mainly on past data, since those are the

most reliable source of unbiased information. Applying econometric models will then lead to

results, which can be tested for their stability and for reliability, especially when compared to

actual data. This procedure will be presented during the following paragraphs with the help of

an example, namely the number of cars produced in Germany every month. I chose this data

set out of two reasons. Firstly, these industry is one of the most important industries within

the German economy, and secondy, my professional engagement with a car-producing

company provides my with some insight into the industy.

In order to generate forecasts, the model will build upon systematic components, such as

trends, seasonality, and cyclicality. Furthermore, concepts like moving averages and auto-

regressions will be incorporated into the model. Following, the data will be tested, whether it

returns to it mean after experiencing a shock. This will be done via a unit-root test. In case of

its occurrence, the unit-root will be removed by a stochastic model. Next, a second series will

be introduced, the collection of leading indicators for the German economy, and incorporated.

According tests towards the causal relationship will be provided and a forecast will be

suggested on the basis of a multivariate model. Concluding, the two forecasts will be

compared.

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2. Introduction of the Series

The automotive industry is one of the most important in the German economy, which can

easily be seen on the number of domestic brands, such as Mercedes-Benz, Audi, Volkswagen,

and BMW. But due to the globalization, those formerly German companies turned into inter-

national, if not global enterprises. Therefore, some of the firms’ products are assembled

abroad, while foreign companies (e.g. Ford) produce now within German borders. Thus the

presented series encorporated the cars produced in Germany.

By looking at the data, one can see that the production of cars increased during the last

four centuries. The

600000

500000

400000

Number of Cars 300000

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0

60

should be underlined

with some scien-

Figure 2

tific evidence. As

can be seen from

figure 2,

distribution seems

to be normal.

Firstly, the mean,

the median, and

the mode do not

differ too much

from each other,

indicating a

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normal distribution. Secondly, the skewness coefficient of –0.076477 underlines this finding.

Thirdly, the kurtosis value of 2.261076 indicates a little flatter tails then normal. Finally, the

Jarque-Bera test examines the hypothesis of independent normally distributed observation.

The reported probability rejects it in favor of the alternative hypothesis. This leads to the

conclusion that the data provide a good basis for an analytical forecast.

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3. Modelling a Univariate Model

3.1 Trend Analysis

Most data series exhibit a trend. Underlying this trend is usually some kind of growth,

such as inflation, population growth, or increases in wealth. But these trend do not always

have to be linear. Most stock market indices increased exponentially, while learning curves

are not seldomly u-shaped (one would speak here of a quadratic trend).

As stated before, the data at hand also seems to follow a trend. This section will be used to

test the data for the occurrence of a linear, quadratic, exponential, or polynomial trend. But

before, the underlying statistics of these models will be introduced:

T t = β 0 + β 1 *TREND t

Linear trend regression 1 :

Quadratic trend regression 2 :

Exponential trend regression 3 : ln(T t ) = ln(β 0 ) + β 1 *TREND t

Polynomial trend regression 4 : T t = β 0 + β 1 *TREND t + β 2 *TREND t ² + β 3 *TREND t ³ +

β 4 *TREND t 4 + β 5 *TREND t

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These four models all weigh the trend differently. The linear trend regression fits a straight

line through the data, where the slope of the line is the estimated growth rate. The other three

methods are non-linear.

Comparing the models will be done by looking at the mean squared error (MSE). This

measure will be optimal, the smaller its value will be. There are different concepts, which take

the MSE into account, namely the R², the Akaike info criterion (AIC), and the Schwarz

criterion (SIC). They differ in the penalizing factor. Generaly, one can say, that the SIC

penalizes the MSE most strongly and will therefore lead to the most reliable indicator, leading

to a more parsimonious (simple) model.

Table 1: Model Selection Criteria

1 See Diebold, p. 72

2 Ibid. p. 75

3 Ibid. p. 78

4 Ibid. p. 76

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The comparison of the results can be seen in Table 1. The first remarkable fact is that all

indicators do not vary much from each other. This can also be seen in the graphs, which are

exhibited in Appendix A. The exponential trend regression cannot be compared this way,

since it relies

on a logarithm

of the data.

Following the

Schwarz cri-

terion, as ar-

gued above,

the quadratic

trend model

(see figure 3)

will be chosen

as the best

model.

Another way of deciding, which model to take, is to test the prediction capabilities of the

several models. The history, forecast and realization of the above suggested models is shown

in the figures five till eight of appendix A. Suprisingly, the forecasts suggest that either the

linear or the exponential trend model is the most accurate (see figure 4). The quardatic trend

regression does not capture the continuing growth in production. This leads to the conclusion

that a linear trend model will probably be the most useful for further analysis.

Figure 4: German Car Production,

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History, Forecast and Realization

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3.2 Seasonal Analysis

Car registrations are highly seasonal, since the purchase of a car is quite an investment. So

it is quite logical that little cars will be registered in december. In january of 2002, a car of

2001 will be counted as being a year old, no matter, in which month in 2001 it was registered.

On the other hand, spring is known to be the time, where most cars are sold. The purchasing

behavior of prospective customers is very likely to be incorporated, when production targets

are set, in order to prevent high inventory levels. Therefore, it is very likely to find seasonality

in the series at hand. As predicted, the series really shows a high seasonality. All of the twelve

monthly dummies are significant at the 5%-level (see Appendix B).

Since the seasons have an influence on car sales, it might be interesting to test for

seasonality by regressing the linear trend model on three quaterly dummies, taking one season

as the base. Since the sales are very likely to lag one month behind the production, the seasons

of interst would be february till april, may till july, and august till october, taking november

till january as the base. The results can be seen in appendix B. AIC and SIC go up compared

to the monthly model and the dummies are not all significant at the 5%-level anymore.

Therefore, the model adjusting for monthly seasonality is superior to the quarterly model and

will consequently be used for further models.

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