Forecasting based on econometric models (time series analysis)

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Forecasting based on econometric models (time series analysis)

A.N. Sakhanova, doctor of economics, professor

Y.Zh. Akhmer, graduate student

Kazakh Ablai Khan university of international

relations and world languages JSC

(Kazakhstan, Almaty)

Abstract. Economic-mathematical modeling is a crucial part of any research in the field of economy. The rapid development of mathematical analysis, operations research, probability theory and mathematical statistics contributed to the formation of various kinds of models of the economy. The problem of studying the relationship of economic indicators is one of the most important in the economic analysis. This issue is central in econometric and it is solved by construction of econometric model and the determination of possibilities of its use for the description, analysis and prediction of real economic processes. This article describes the forecasting and its classification. The method of time series is described in more detail and the real example of simulation of exchange ratio KZT/RUB between 2006-2013 is given. The main stages of the work are analyzed.

Keywords: modelling, forecasting, time series analysis, prediction, econometric, methods.

Прогнозирование на основе эконометрических моделей (анализ временных рядов)

Аннотация. Экономико-математическое моделирование является важной частью любого исследования в области экономики. Быстрое развитие математического анализа, исследования операций, теории вероятностей и математической статистики способствовало формированию различного рода моделей экономики. Проблема изучения взаимосвязи экономических показателей является одним из наиболее важных в экономическом анализе. Этот вопрос занимает центральное место в эконометрических и она решается путем построения эконометрической модели и определения возможностей его использования для описания, анализа и прогнозирования реальных экономических процессов. В данной статье приводится описание прогнозирование и его классификации. Более подробно описан метод временных рядов и приведен реальный пример моделирования валютного соотношения тенге к рублю за 2006-2013. Основные этапы работы проанализированы.

Ключевые слова: моделирование, прогнозирование, анализ временных рядов, прогнозирование, эконометрические методы.

econometric model indicator exchange rate

Introduction

Forecasting is activity, which aimed to identify and study possible alternatives to the future development of the company. Main role is given to forecasting product sales. The main purpose of the forecast is to determine the trend factors affecting market conditions. Forecasting can be the short-term forecasts, they are generally made for 1 -- 1.5 years, medium-term for 4-6 years and long-term -- 10-15 years. The main emphasis in the short-term forecasting is done on a quantitative and qualitative assessment of changes in the volume of production, supply and demand, the level of competitiveness of the goods and price indices, exchange rates, currency ratios and credit conditions. Medium-term and long-term forecasting system based on forecasts of the market conditions, supply and demand, for the protection of the environment constraints, international trade. As a forecasting tool used formalized quantitative methods (factor, statistical analysis, mathematical modeling), methods of expert estimates based on experience and intuition of experts on this product and the market.[2]

In the analysis of economic phenomena on the basis of economic and mathematical methods occupy a special place model, revealing the quantitative relation between the studied parameters and factors influencing them. Scientific disciplines, the subject of which is to study the quantitative aspects of economic phenomena and processes by means of mathematical and statistical analysis, is econometrics, in which the results of the theoretical analysis of the economy are synthesized with the conclusions of mathematics and statistics. The main objective of econometrics is to test economic theories on factual material using methods of mathematical statistics [3].

As part of econometrics there is scientific and educational discipline «Mathematical methods of forecasting.» Its purpose is the development, research and application of modern mathematical methods of econometric prediction of socio-economic phenomena and processes, and methods should be worked out to a level allowing them to use the economist in practice, engineers and managers. The main objectives of this discipline are the development, research and application of advanced mathematical and statistical forecasting methods. The theoretical basis of forecasting methods are mathematical disciplines (especially the theory of probability and mathematical statistics, discrete mathematics, operations research), as well as economics, economic statistics, management, sociology, political science and other social and economic sciences.

Classification of methods of forecasting

Model prediction is a functional representation of adequately describing the process under study is the basis for its future value. Now it is accepted to use the English abbreviations of names as the models and methods. For example, there is a famous model of auto regression of integrated moving average, taking into account external factors (auto regression integrated moving average extended, ARIMAX). The model and the corresponding method are usually called ARIMAX, and sometimes model (method) Box-Jenkins on behalf of the sponsors [1].

The term «forecasting method» is much broader concept of «forecasting model». In this regard, in the first stage classification methods are generally divided into two categories: intuitive and formalized.

Intuitive forecasting methods deal with judgments and estimates of experts. Today, they are often used in marketing, economics, politics, because the system, which is necessary to predict the behavior, which is very complex and not amenable to mathematical description, or is very simple and does not require such a description.

Formalized methods are forecasting methods, which resulted in building a prediction model, define a mathematical relationship, which allows you to calculate the future value of the process, that is, to make a prediction. It is necessary to proceed to the classification of forecasting models. In the first stage model should be divided into two groups: the domain models and time series models.

Domain models are mathematical models of prediction, for the construction of which use domain laws. For example, the model on which make the weather forecast contains equations of fluid dynamics and thermodynamics. Forecast of development of the population is on the model based on differential equations. The forecast level of a person's blood sugar, diabetic, is based on a system of differential equations. In short, these models are used depending inherent in a particular subject area. This kind of model is peculiar to an individual approach in the design.

Time series models are mathematical models predict that seek to find the dependence of the future values of the past in the process and, depending on this to calculate the forecast. These models are versatile for a variety of subject areas, that is, their general appearance does not change depending on the nature of the time series. We can use neural networks to predict the air temperature, and after a similar model to apply neural networks for the prediction of stock market indices [5].

Time series analysis

Now we will consider more precisely this method. Time series analysis is a statistical technique that deals with time series data, or trend analysis. Time series data means that data is in a series of particular time periods or intervals. The data is considered in three types:

Time series data: A set of observations on the values that a variable takes at different times.

Cross-sectional data: Data of one or more variables, collected at the same point in time.

Pooled data: A combination of time series data and cross-sectional data.[4]

Time series analysis is one of the main tools of observing collected data over time with sequential nature and predict its future values.

Example of Time Series Analysis:

As an example of time series analysis let's try to apply it to exchange rate between KZT and RUB between 2006-2013. We will develop ARMA model to forecast this process and in order to be sure that our forecast is true, we will divide data set into training set and test set.

Analysis

As already mentioned we divide data into two parts. This data includes 96 months from 2006 to 2013. The training set will consist of 80 months and the rest 16 is test set. We will use GRETL software to do analysis.

Figure 1. Time series plot of monthly values of exchange rate between KZT and RUB

Figure 2. Correlogram of training set

The first figure shows us, that our process does not behave as a stationary process. And also from correlogram, we cannot be sure about stationarity, because many bars crossing the confidence bands. ARIMA(p,q,d) models are, in theory, the most general class of models for forecasting a time series which can be made to be “stationary” by differencing (if necessary). Random variable, that is a time series, is stationary if its statistical properties are all constant over time [6].

Figure 3. Time series plot of the first difference of training set

Now, from figure 3 we can notice that data after the first difference of the training set seem to be stationary. Nevertheless, we want to be sure, so it leads to use one of the tests, to check whether it is stationary or not. This test is called Augmented Dickey-Fuller (ADF) test.

Results obtained from test are

test statistic: tau_c(1) = -8.82741

p-value 1.715e-007

Our asymptotic value is very close to zero. According to our test statitstic and p-value with 97% confidence we are sure that process is stationary.

Modelling:

Our ARIMA(p,q,d) model consists of three parameters, but we already defined d as 1, because we used only the first difference of our time series. One of the ways to identify p and q is to look at SACF and SPACF, and observe which lag crosses the confidence bands. In this case, SACF and SPACF is never crossed, hence we need to investigate different pairs of parameters and consider Akaike information criterion (AIC) and the Bayes information criterion (BIC, also known as Schwarz criterion).

Table 1. AIC/BIC values for different parameters

Note: According to AIC/BIC values it makes sense to take parameters (1,1)

Forecasting

As mentioned before, this is one of the main purposes of time series analysis and modeling, and any kind of modeling in general. The models that were created with the training set now will be used to predict what should be the next values of the plots. It can be seen in figures 4. In all of these plots (in blue) the model makes a “prediction” of the past, i.e. it determines what would have been the value if the DGP is exactly the same as the model. Then, it compares that with the actual previous values (red lines). The green dots represent the forecasts for the ”future” (with respect to the training set, not the actual future), which are compared with the test set values.

Figure 4. The forecast with ARMA(1,1)

Results

Forecast evaluation statistics Value

Mean Error --0.06793

Root Mean Squared Error 0.16916

Mean Absolute Error 0.14093

Mean Percentage Error -1.5005

Mean Absolute Percentage Error 2.9799

Theil's U 1.6124

Bias proportion, UM 0.16125

Regression proportion, UR 0.27067

Disturbance proportion, UD 0.56808

Coefficient std. error z p-value

const 0.00244696 0.00349228 0.7007 0.4835

phi_1 0.780132 0.0749261 10.41 2.19e-025 ***

theta_1 ?1.00000 0.0325462 ?30.73 2.60e-207 ***

According to results we chose parameters with the best option. So the models were used to forecast values which were compared with those of the test set. Based on the smallest mean squared error (0.16916), we conclude the best models, which describes our time series is ARMA(1,1).

As we can see forecasting is not simple thing, and the normal ARMA model gives us the whole picture of process, but at the same time, it is too simple, and cannot give exact forecast.

Conclusion

Every economic research related to the analysis of empirical data, as a rule, correspond to their econometric models. A striking example of the use of econometric methods is the analysis of the dynamics of prices and quality of life. Almost any area of the economy has to do with the statistical analysis of empirical data, and therefore has a certain econometric techniques in their toolbox.

Using econometric techniques should be evaluated different values and dependencies. Econometric methods are effective tools in the work of the manager and the engineer in charge of the specific problems, designed for the analysis of statistical data and econometric models of building concrete economic and techno-economic phenomena and processes. So that, the importance of using econometric methods in forecasting in general are showed. The classification of forecasting was introduced and the full modelling process using time series analysis was described for particular example. All analysis, computations, modelling and forecasting were done using software GRETL.

References

1. Tikhonov E.E. Predicting the market conditions. Nevinnomyssk, 2006. 221

2. Lotov A.V. Introduction to the economic and mathematical modeling. M Science, 1984.

3. E. Jantsch, Forecasting of scientific and technical progress. / Transl. from English, Progress, 1974.

4. Box, G.E.P. and G.M. Jenkins Time Series Analysis, Forecasting and Control. Revised Edition. Holden Day, SanFrancisco, 1976.

5. Shumway, R. H. and D. S. Stoer Time Series Analysis and Its Applications: With R Examples, Springer, 2010.

6. Brockwell P.J. and R.A. Davis, Introduction to Time Series and Forecasting, 2nd Edition, Springer, 2002.

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