# What are quantitative forecasting techniques

## Forecasting technology

The entirety of the qualitative and quantitative methods that are used to develop a forecast. Often one understands by this the numerical solution procedure for the evaluation of quantitative - prognosis models.

provides methods to predict the development of an economic variable y as a function of time (t, as an independent variable). A distinction is made between · univariable methods which only consider the time factor as an influencing variable; · Multivariable methods that take into account other economic variables in addition to time. The univariable forecast (discussed here alone) should be based on a previously created time series. This tries to derive connections between a variable and time from past observations. The prognosis principle is then based on transferring these relationships directly or in a meaningfully modified form to the future. In principle, futuristic methods (without deriving from regularities) or heuristic (based on experience) methods are also possible when creating forecasts. For economics, however, deterministic (possibly with a stochastic remainder) and stochastic methods are more common. An additional subdivision of the forecasts is based on their maturity. A distinction is made between short-term (up to a year), medium and long-term forecasts. Which forecast options are to be selected depends on the one hand on the structure of the available data (possibility of extrapolation into the future) and on the other hand on the given objective (forecast horizon). a) Deterministic methods:

1. As methods for short-term forecasts (also with a multivariable view), methods of regression analysis are available, whereby time is regarded as an independent variable and the variable to be forecast as a dependent variable. A functional relationship from the past is transferred to the future. The corresponding parameters are obtained using the least sums of squares method. This results in forecast values (y) of the variables to be forecast in the form of a polynomial

or in the form of exponential equations in each case for the prognosis time t. In principle, however, all possible functional relationships are conceivable here. Another method is based on the principle of moving averages. The special structure of a time series, which cannot always be represented in the form of a simple function, is taken into account. A time series is usually made up of a smooth component, which represents the long-term development (trend), and a seasonal component, which records regularly recurring cyclical fluctuations. Therefore, different methods are needed to capture these two phenomena, i.e. to eliminate the irregular fluctuations and the stochastic residues. The moving average method determines the trend component; this is followed by an estimate of the seasonal figure. In order to determine a (linear) trend, the arithmetic mean of the last m periods is formed

as well as the arithmetic mean of these moving averages, also from the last m periods (moving averages

2nd order),

The method of weighted moving averages is then better suited as the forecast value for the time t + i. The values included in the averaging are weighted according to their importance, i.e. more recent data are included in the calculation more than older ones. This ultimately leads to the principle of exponential smoothing, which is a special form of the weighted moving averages method. The past values are no longer directly included in the calculation, but only in the form of a parameter a, the so-called smoothing parameter, which must be suitably selected in accordance with the available data. The following applies here: 0 S a <_

1. The period t is assigned as an exponentially smoothed mean

For the slope of the trend straight line results

and thus follows as the forecast value for the period t + i

Here, however, the problem arises of finding suitable initial values.

2. Here, too, regression analysis methods come into consideration as methods for medium and long-term prognoses, as long as no saturation level has to be considered for the variable to be prognosticated. Esp. In the case of growth forecasts (e.g. new demand forecasts), there will be an asymptotic course against the saturation level. Here it is useful and necessary to make forecasts with the help of the classes of logistic or GOMPERTZ functions that take this behavior into account. The functions considered must have a first derivative not equal to zero, the so-called growth rate. The antiderivative of the class of logistic functions is with A. as the saturation level and G (t) = fg (t) dt with g (t) as a description of the behavior (e.g. buying behavior on the market). Depending on the design of g (t) and thus of G (t), a symmetrical or asymmetrical curve shape is obtained for f. B) Stochastic methods:

The basic principle of forecasts (discussed here alone) using MARKOV processes assumes that the variable to be forecast is a random variable that has a given vector of the initial probabilities (ao) of its characteristics at the time to = 0. Assuming that the number of occurrences is finite and the same at each point in time, the vector of the probabilities results at the point in time t = 1 P denotes the matrix of the transition probabilities from point in time t-1 to point in time t. This transition scheme itself is not a function of time. Therefore also applies

Thus one can predict probabilities for every point in time t Literature: Lewandowski, R. (1980). Kendall, M., Stuart, A. (1976). Mertens, P. (1973)

provides methods to predict the development of an economic variable y as a function of time (t, as an independent variable). A distinction is made between · univariable methods which only consider the time factor as an influencing variable; · Multivariable methods that take into account other economic variables in addition to time. The univariable forecast (discussed here alone) should be based on a previously created time series. This tries to derive connections between a variable and time from past observations. The prognosis principle is then based on transferring these relationships directly or in a meaningfully modified form to the future. In principle, futuristic methods (without deriving from regularities) or heuristic (based on experience) methods are also possible when creating forecasts. For economics, however, deterministic (possibly with a stochastic remainder) and stochastic methods are more common. An additional subdivision of the forecasts is based on their maturity. A distinction is made between short-term (up to a year), medium and long-term forecasts. Which forecast options are to be selected depends on the one hand on the structure of the available data (possibility of extrapolation into the future) and on the other hand on the given objective (forecast horizon). a) Deterministic methods:

1. As methods for short-term forecasts (also with a multivariable view), methods of regression analysis are available, whereby time is regarded as an independent variable and the variable to be forecast as a dependent variable. A functional relationship from the past is transferred to the future. The corresponding parameters are obtained using the least sums of squares method. This results in forecast values (y) of the variables to be forecast in the form of a polynomial

or in the form of exponential equations in each case for the prognosis time t. In principle, however, all possible functional relationships are conceivable here. Another method is based on the principle of moving averages. The special structure of a time series, which cannot always be represented in the form of a simple function, is taken into account. A time series is usually made up of a smooth component, which represents the long-term development (trend), and a seasonal component, which records regularly recurring cyclical fluctuations. Therefore, different methods are needed to capture these two phenomena, i.e. to eliminate the irregular fluctuations and the stochastic residues. The moving average method determines the trend component; this is followed by an estimate of the seasonal figure. In order to determine a (linear) trend, the arithmetic mean of the last m periods is formed

as well as the arithmetic mean of these moving averages, also from the last m periods (moving averages

2nd order),

The method of weighted moving averages is then better suited as the forecast value for the time t + i. The values included in the averaging are weighted according to their importance, i.e. more recent data are included in the calculation more than older ones. This ultimately leads to the principle of exponential smoothing, which is a special form of the weighted moving averages method. The past values are no longer directly included in the calculation, but only in the form of a parameter a, the so-called smoothing parameter, which must be suitably selected in accordance with the available data. The following applies here: 0 S a <_

1. The period t is assigned as an exponentially smoothed mean

For the slope of the trend straight line results

and thus follows as the forecast value for the period t + i

Here, however, the problem arises of finding suitable initial values.

2. Here, too, regression analysis methods come into consideration as methods for medium and long-term prognoses, as long as no saturation level has to be considered for the variable to be prognosticated. Esp. In the case of growth forecasts (e.g. new demand forecasts), there will be an asymptotic course against the saturation level. Here it is useful and necessary to make forecasts with the help of the classes of logistic or GOMPERTZ functions that take this behavior into account. The functions considered must have a first derivative not equal to zero, the so-called growth rate. The antiderivative of the class of logistic functions is with A. as the saturation level and G (t) = fg (t) dt with g (t) as a description of the behavior (e.g. buying behavior on the market). Depending on the design of g (t) and thus of G (t), a symmetrical or asymmetrical curve shape is obtained for f. B) Stochastic methods:

The basic principle of forecasts (discussed here alone) using MARKOV processes assumes that the variable to be forecast is a random variable that has a given vector of the initial probabilities (ao) of its characteristics at the time to = 0. Assuming that the number of occurrences is finite and the same at each point in time, the vector of the probabilities results at the point in time t = 1 P denotes the matrix of the transition probabilities from point in time t-1 to point in time t. This transition scheme itself is not a function of time. Therefore also applies

Thus one can predict probabilities for every point in time t Literature: Lewandowski, R. (1980). Kendall, M., Stuart, A. (1976). Mertens, P. (1973)

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