use of arima and var models for exchange rate forecasting ...

Romanian Statistical Review - Supplement nr. 1 / 202066

USE OF ARIMA AND VAR MODELS FOR EXCHANGE RATE FORECASTING AND

ACCURACY TESTINGAssoc. prof. Mădălina-Gabriela ANGHEL PhD ([email protected])„Artifex” University of Bucharest

Abstract In this article, the author focused on presenting multivariate models. These multivariate models use the decomposition of the considered course, providing information that has essentially quantification possibilities. In this way, the autoregressive vector methodology called VAR is a method found in the analysis of time series, flexibility and adaptation to these types of analyzes. These variables are the basis of the construction, the realization of equations that by partial derivation provide a system of equations by which we can easily identify the parameters according to which we can estimate the phenomenon under investigation, in our case, the exchange rate. In order to take into account the results obtained by these multivariate methods, even the univariate ones, it is necessary to apply accuracy tests of the forecasts. In order to determine the performance of the different forecasting methods, a series of indicators can be used, these measuring the distance between the actual, historical values of the respective series and the values obtained from the resolution, the use of the respective models. A number of indicators, such as the RMSE, show how the exchange rate will be able to evolve over a given period of time. The tests that are applied are statistical tests that highlight the accuracy with which the results obtained are edifying from the manager’s point of view at the macroeconomic level or at the microeconomic level. The models presented and used on the basis of data for example, show that the accuracy tests are positive and from this point of view the calculated, estimated indicators are reliable, have a high probability and thus ensure the completion of the macroeconomic forecasts we are talking about by considering the trend that the exchange rate will have in a future period. Keywords: models, accuracy tests, forecast, exchange rate, indicators. JEL classification: C10, E10, G10

Revista Română de Statistică - Supliment nr. 1 / 2020 67

Introduction In this article Using ARIMA and VAR models for exchange rate forecasting and accuracy testing, the author started from the need to approach methodologically and then and, by way of example, how multivariate models can be used in forecasting. exchange rate. The main models and data are presented in relation to the variables underlying the construction of exchange rate forecast models. These variables are those that need to be correlated and have a concerted influence on the overall future evolution of a country’s economy. The exchange rate forecast models are exemplified by a careful study of how the relationship between the dollar and euro evolves over a period of time, as it prefigures and bases the relations that can be established in the field of international economic and technical-scientific exchanges. between the member countries of the European Union and the United States. In this sense, the example presented is edifying and it can be extended with good results in the analysis of the exchange rate forecast between any national currency and a freely convertible currency to which imports and exports are made. After clarifying these aspects, a brief presentation of the exchange rate accuracy tests is made. Of course, in the activity of managing the economy of a country, a complex forecast is needed, but that is also based on the sectoral forecasts that have been made so that the possibility of influence of each identified factor (statistical variable) can be known, in order to one could take the necessary measures. From this point of view, we find that it is sufficient to apply some statistical tests that will tell us whether the calculated indicators and if the methodology used satisfies the forecast. In exchange for a poor forecast regarding the exchange rate, because we refer to it, it is often better to give up than to complicate the data. Only, in order to show the contribution of international economic relations to the economic results obtained by a state, by a country, it is necessary to make this forecast of the exchange rate. In this article, the example used is accompanied by a series of graphs that show how it behaves in terms of the tests used, models such as ARIMA or VAR, used in the exchange rate forecast.

Literature review Anghelache Constantin, Anghel Mădălina Gabriela (2016) approaches in their work the various theoretical and practical concepts of statistics. Anghelache Constantin, Partachi Ion, Anghel Mădălina-Gabriela (2017) develops a predictive analysis on economic growth. Cheung, Yin-Wong, Menzie D. Chinn, şi Antonio Garcia Pascual (2005) analyzes from a statistical-econometric point of view the evolution of the exchange rate


since the 1990s. Driver, Rebeca L., P. F. Westaway (2004) addresses various concepts that underpin the evolution of exchange rates. Gutierrez, Carrasco, Enrique, Carlos, Reinaldo Castro Souza şi Osmani Teixeira de Carvalho Guielen (2007), approaches optimal Lag length selection in VAR models cointegrated with poor shape of common cyclic characteristics. Iacob, S,.V. (2019) approaches various statistical-econometric methods for modeling economic phenomena. MacDonald, Ronald (2000) presents an overview of methods for calculating equilibrium exchange rates.

Methodology, data, discussions, results • Accuracy testing of forecasts Different indicators can be used to determine the performance of the different forecast models, by measuring the distance between the actual, historical values of the target series and the values obtained from the different models used. The indicators can be grouped, taking into account different criteria, in indicators that change their scale with the series level or not. Another classification criterion is related to the way they are calculated, identifying here indicators based on the square or absolute deviation depending on how the distance between the actual and the predicted values is measured. Also, the indicators can be classified into quantitative and qualitative indicators, the latter characterizing the model used from the point of view of the type of errors it generates. A first indicator addressed in this analysis is RMSE (root mean of squared error, that is radically from the average of the squares of forecast errors), being determined with the relation:

(1)

3

de vedere statistico-econometric evoluția cursului de schimb din anii 1990. Driver, Rebeca L.,

P. F. Westaway (2004) abordează diverse concepte care stau la baza evoluției ratelor de

schimb. Gutierrez, Carrasco, Enrique, Carlos, Reinaldo Castro Souza şi Osmani Teixeira de

Carvalho Guielen (2007), abordează selecția lungimii Lag optimă în modele VAR cointegrate

cu o formă slabă a caracteristicilor ciclice comune. Iacob, S,.V. (2019) abordează diverse

metode statistic-econometrice de modelare a fenomenelor economice. MacDonald, Ronald

(2000) prezintă o imagine de ansamblu a metodelor pentru calcularea ratelor de schimb de

echilibru.

Metodologie, date, discuții, rezultate

Teste de acurateţe a prognozelor

Pentru a determina performanţa diferitelor modele de prognoză se pot utiliza diferiţi

indicatori, aceştia măsurând distanţa între valorile reale, istorice, ale seriei vizate şi valorile

obţinute din diferitele modele utilizate.

Indicatorii pot fi grupaţi, având în vedere diferite criterii, în indicatori care îşi modifică

scala odată cu nivelul seriei sau nu. Un alt criteriu de clasificare este legat de modul de calcul

al acestora, identificând aici indicatori bazaţi pe abaterea pătratică sau absolută în funcţie de

modul în care este măsurată distanţa dintre valorile reale şi cele prognozate. De asemenea,

indicatorii pot fi clasificaţi în indicatori cantitativi şi calitativi, cei din urmă caracterizând

modelul folosit din punctul de vedere al tipului de erori pe care-1 generează.

Un prim indicator abordat în această analiză este RMSE (root mean of squared error,

adică radical din media pătratelor erorilor de prognoză), fiind determinat cu relația:

= ∑ (1)

Indicatorul îşi modifică scala odată cu scala variabilei investigate, neputându-se

compara decât rezultatele unor modele aplicate asupra aceleiaşi variabile. Este un indicator

cantitativ, iar descompunerea sa conducând la determinarea altor indicatori de acelaşi tip.

Indicatorul MSFE (mean squared of forecast error, adică media pătratelor erorilor de

prognoză), care reprezintă pătratul RMSE, poate fi descompus în părţi corespunzătoare

abaterii mediei prognozelor de la media seriei reale, o parte corespunzătoare deplasării

varianţei şi o parte ce provine din erorile nesistematice ale modelului. Astfel, pornind de la

indicatorul Theil se poate defini indicatorii U- medie (definit şi ca „ bias proportion ”), U-var

The indicator changes its scale with the scale of the investigated variable, and can only compare the results of some models applied on the same variable. It is a quantitative indicator, and its decomposition leads to the determination of other indicators of the same type. The mean squared of forecast error, which is the mean of the squares of forecast errors, which represents the RMSE square, can be broken down into parts corresponding to the deviation of the average of the forecasts from the average of the real series, a part corresponding to the variance shift and a part that comes from the non-systematic errors. of the model. Thus, starting from the Theil indicator one can define the indicators U-average (also defined as „bias proportion”), U-var (also defined as „variance proportion”) and U-cov


(„covariance proportion”). The formulas used for determining these indicators are:

4

(definit şi ca „ variance proportion ”) şi U-cov („covariance proportion"). Formulele folosite

pentru determinarea acestor indicatori sunt:

= ∑ (2)

= − + − + 21 − (3)

= (4)

= (5)

= (6)

Valorile ultimilor trei indicatori se constituie în criterii de evaluarea a performanţelor

modelelor testare, distribuţia ponderilor acestora, datorită faptului că suma acestor trei

indicatori este 1. Astfel, este permisă caracterizarea modelului din punctul de vedere al

ponderii din eroarea totală explicată de abaterea mediei, de cea a varianţei şi de factori

nesistematici. Desigur, performanţa unui model este cu atât mai mare, cu cât ponderea

ultimului indicator este mai mare în ansamblul celor trei, într-un anumit sens distribuţia

constituindu-se într-un criteriu de evaluare din punct de vedere calitativ al modelului.

Aceşti indicatori permit compararea diverselor prognoze din punctul de vedere al

modului în care surprind media şi varianţa seriei prognozate, fiind criterii secundare de

apreciere a unui model, cel principal fiind constituit de indicatori ai deplasării prognozelor

acestuia de la valorile istorice.

Un alt indicator al performanţei în prognoză se constituie în MAE (mean absolute error,

adică media valorilor în termeni absoluţi a erorilor). Acest indicator îşi modifică scala odată

cu scala variabilei asupra căreia este aplicat. Prin urmare, indicatorul poate fi ajustat pentru a

deveni invariant din punctul de vedere al scalei variabilei, obţinându-se în acest fel MAPE.

Valorile acestui indicator pornesc de la 0, această valoare indicând o suprapunere perfectă

între prognoză şi variabila prognozată, iar valori mai mari indică o depărtare a acesteia de

The values of the last three indicators are constituted by criteria for evaluating the performance of the test models, the distribution of their weights, due to the fact that the sum of these three indicators is 1. Thus, it is allowed to characterize the model from the point of view of the total error explained by the deviation of the average, by that of variance and non-systematic factors. Of course, the performance of a model is the higher, as the weight of the last indicator is higher in all three, in a certain sense the distribution constituting a qualitative evaluation criterion of the model. These indicators allow the comparison of the various forecasts from the point of view of the way in which they surprise the average and the variance of the forecasted series, being secondary criteria of appreciation of a model, the main one being constituted by indicators of the movement of its forecasts from the historical values. Another indicator of the performance in the forecast is the MAE (mean absolute error, ie the average of the values in absolute terms of the errors). This indicator changes its scale with the scale of the variable to which it is applied. Therefore, the indicator can be adjusted to become invariant from a variable scale point of view, thus obtaining MAPE. The values of this indicator start from 0, this value indicating a perfect overlap between the forecast and the predicted variable, and higher values indicate a distance of it from the variable whose properties try to replicate them. The relations for the MAE and MAPE indicators are:


5

variabila ale cărei proprietăţi încearcă să le replice. Relațiile pentru indicatorii MAE și MAPE

sunt:

= ∑ || (7)

= ∑ (8)

Un alt indicator utilizat pentru caracterizarea prognozelor realizate cu diferite tipuri de

modele este Theil IC („Theil Inequality Coefficient"). Acest indicator este invariant din

punctul de vedere al scalei variabilei asupra căreia este aplicat, putând fi utilizat pentru

compararea unor modele aplicate pe economii diferite. Valorile acestuia sunt cuprinse între 0

şi 1, prima valoare indicând maximul de performanţă în prognoză, adică suprapunerea

acesteia cu seria istorică, pe eşantionul analizat, iar cea de a doua valoare, lipsa de legătură a

modelului utilizat cu seria istorică folosită. Având în vedere că Theil IC se bazează pe MSFE,

se poate realiza o descompunere a acestuia în cei trei factori descrişi anterior (medie, varianţă,

factori nesistematici), utilizând relația:

ℎ = ∑ ∑ ∑

(9)

În figura următoare este prezentată evoluția acestui indicator pentru patru modele.

Figura 1: Evoluția indicatorului Theil IC

(7)

5


sunt:

= ∑ || (7)

= ∑ (8)










ℎ = ∑ ∑ ∑

(9)



(8)

Another indicator used to characterize forecasts made with different types of models is Theil IC („Theil Inequality Coefficient”). Its values are between 0 and 1, the first value indicating the maximum performance in the forecast, that is, its overlap with the historical series, on the analyzed sample, and the second value, the lack of connection of the model used with the historical series used. Given that Theil IC is based on MSFE, it can be decomposed into the three factors described above (mean, variance, non-systematic factors), using the relation:

5


sunt:

= ∑ || (7)

= ∑ (8)










ℎ = ∑ ∑ ∑

(9)



(9)

The following figure shows the evolution of this indicator for four models.

Theil IC indicator evolutionFigure 1


To characterize the models used in the paper, indicators of their approximation to the historical series and the decomposition of the mean of the square of errors into the corresponding part of the mean, variance and non-systematic factors will be used as indicators of the quality of the forecasts made. Comparing the way in which the characterization of the evolution of the forecasts is realized, it is obvious that the two big types of indicators, those based on the average square deviation and those based on the absolute average deviation, have advantages and disadvantages, because the average square deviation can distort the results. Given this disadvantage of the indicators based on the quadratic mean, but also the advantage of having a transparent decomposition of the forecast errors in parts corresponding to the mean, the variance and the non-systematic factors, an evaluation and presentation of all the indicators, regardless of the calculation method, is desirable. , this being likely to lead to significant conclusions regarding the chosen measurement method. All these indicators will be applied on a cross-section of the forecasts. Thus, to characterize the properties in the forecast at 12 months away from the initial moment, the distance between the historical series and all the forecasts at 12 months away from the initial moment will be analyzed, the number of forecasts being the same as the values of the historical series (the forecasts being made from all points in the historical series).

• ARIMA and VAR models for the eur / usd exchange rate forecast This study aims to select a model with both univariate and multivariate models that differ from the former by the variables included to achieve a forecast of the eur / usd exchange rate. In the first case, an estimation of the eur / usd exchange rate was made based on 6 ARIMA models, not before this time series was subjected to the general transformations imposed by the univariate models (Augmented Dickey-Fuller, Philips stationarity tests). Platform). In the second case, an estimate of the eur / usd exchange rate was made based on nine VAR models, and these were tested for the number of lags (between 1 and 10) and for stability. In both cases, the estimation took place on the set interval, using the econometric program EViews. Based on these univariate and multivariate models estimated on the established sample, we tried to make a forecast inside the sample over the respective interval. A forecast was made starting from each point of the sample mentioned above, all forecasts being processed then, in a cross section and grouped on forecast horizons. The forecasting method used indicates how


relevant the structure of the model is in terms of capturing the dynamics of the chosen variables.For the simplicity of the forecasts presentation, the six ARIMA models and the most efficient six of the nine VAR models analyzed from the point of view of the accuracy of the forecasts were chosen.

27-month eur / usd exchange rate forecasts for univariate modelsFigure 2

7

prognozele fiind prelucrate apoi, într-o secţiune transversală şi grupate pe orizonturi de

prognoză. Metoda de prognoză folosită indică cât de relevantă este structura modelului din

punctul de vedere al capturării dinamicii variabilelor alese.

Pentru simplitatea prezentării prognozelor s-au ales cele șase modele ARIMA și cele

mai performante șase din cele nouă modele VAR analizate din punctul de vedere al preciziei

prognozelor.

Figura 2: Prognozele cursului de schimb eur/usd pe 27 de luni pentru modelele univariate

model RWd model AR1d

model AR1dt model AR1

model ARIMA 2 model ARIMA1


Forecasts of the 27-month eur / usd exchange rate for multivariate models

Figure 3

8

Figura 3: Prognozele cursului de schimb eur/usd pe 27 luni pentru modelele multivariate

model VAR1 model VAR3



Notă: linia îngroșată reprezintă seria istorică, iar celelalte linii reprezintă serile prognozate

Note: the thick line represents the historical series, and the other lines represent the predicted series.


As can be seen from the figures of the forecasts (grouped on the forecast horizons), each having a different starting time, it presents quite visible deviations in the case of the univariate models and they fail to catch the historical series (which happened in reality), due to the fact that Univariate models are simple models that take into account only the past values of the explained variable and the errors recorded in the past.Regarding the forecasts made on the basis of the multivariate models we can notice quite small deviations of the forecasted series compared to the historical series. The performance of these latter models is largely explained by the inclusion of several explanatory variables with varying degrees of influence on the exchange rate.For a more in-depth analysis of these forecast models, the accuracy indicators of the forecast (RMSE, MAE, MAPE and Theil IC) were used, which actually shows the performance of the forecast, ie how far we are with our forecast from the historical series. Based on these indicators we will make a selection so that there remain two univariate models and two multivariate models for comparison, and finally the choice of a single model as the most representative for the economic climate.

Graphs of forecast performance indicators for univariate modelsFigure 4

9

După cum se observă din figurile prognozelor (grupate pe orizonturi de prognoza),

având fiecare un moment de plecare diferit, prezintă abateri destul de vizibile în cazul

modelelor univariate şi nu reuşesc să prindă seria istorică (ce s-a întâmplat în realitate),

datorită faptului ca modele univariate sunt modele simple care iau în calcul numai valorile din

trecut ale variabilei explicate, cât și erorile înregistrate în trecut.

În ceea ce priveşte prognozele realizate pe baza modelelor multivariate putem remarca

abateri destul de mici a seriilor prognozate faţă de seria istorică. Performanţa acestor modele

din urmă este explicată în mare măsură de includerea mai multor variabile explicative cu

diferite grade de influenţă asupra cursului de schimb.

Pentru o analiză mai aprofundă a acestor modele de prognoză s-a apelat la indicatorii de

acurateţe a prognozei (RMSE, MAE, MAPE şi Theil IC ), care ne arată de fapt performanţa

prognozei, adică cât de departe suntem cu prognoza noastră față de seria istorică. Pe baza

acestor indicatori vom face o selecţie astfel încât sa rămână două modele univairate și două

modele multivariate pentru comparaţie, iar în cele din urmă alegerea unui singur model ca

fiind cel mai reprezentativ pentru climatul economic.

Figura 4: Graficele indicatorilor de performanţă în prognoză pentru modelele univariate

MAPE pentru modele univariate MAE pentru modele univariate

RMSE pentru modele univariate MSFE pentru modele univariate


10

Figura 5: Graficele indicatorilor de performanţă în prognoză pentru modelele multivariate

MAE pentru modelele VAR MAPE pentru modelele VAR

RMSE pentru modelele VAR MSFE pentru modelele VAR

Theil IC pentru modelele VAR

Graphs of forecast performance indicators for multivariate modelsFigure 5

10

Figura 5: Graficele indicatorilor de performanţă în prognoză pentru modelele multivariate

MAE pentru modelele VAR MAPE pentru modelele VAR

RMSE pentru modelele VAR MSFE pentru modelele VAR

Theil IC pentru modelele VAR

From the analysis of the graphs, the univariate models ARIMA1,

ARIMA2 and multivariate VAR4, VAR9 were selected following the


comparative analysis based on the performance indices as the most performant models, as they have quite small forecast errors. In order to select the best model out of the four mentioned above, the Theil IC indicator was used - as a quantitative indicator and its decomposition in average, variance and non-systematic factors - as qualitative indicators. It was envisaged to obtain forecasts as far as possible in terms of mean and variance (minimization U - mean and U - var and maximization U - cov), along with the minimization of the quantitative indicator Theil IC. Thus, with this indicator we can measure the error from three points of view. First, part of the error (percentage) comes from the fact that the model does not take the average of the real series, secondly, part of the error comes from the fact that the model does not take the variance, that is, the model overestimates or underestimates the variance and lastly part of the error comes from the factors that the model cannot surprise (non-systematic factors).

Decomposition of Theil IC indicator for comparative modelsFigure 6

11

Din analiza graficelor modelele univariate ARIMA1, ARIMA2 şi multivariate VAR4,

VAR9 au fost selectate în urma analizei comparative pe baza indicilor de performanţă ca fiind

modelele cele mai performante, deoarece prezintă erori în prognoză destul de mici.

În vederea selectării celui mai bun model din cele patru amintite mai sus s-a apelat la

indicatorul Theil IC - ca indicator cantitativ şi descompunerea acestuia în medie, varianţă şi

factori nesistematici - ca indicatori calitativi.

S-a avut în vedere obţinerea unor prognoze pe cât se poate de nedeplasate în termeni de

medie şi varianţă (minimizare U - medie şi U - var şi maximizare U - cov), odată cu

minimizarea indicatorului cantitativ Theil IC. Astfel, cu ajutorul acestui indicator putem

măsura eroarea din trei puncte de vedere. În primul rând o parte din eroare (procent) provine

de la faptul că modelul nu prinde media seriei reale, în al doilea rând o parte din eroare

provine de la faptul că modelul nu prinde varianţa, adică modelul supraestimează sau

subestimează varianţa şi în ultimul rând o parte din eroare provine de la factorii pe care

modelul nu-i poate surprinde (factori nesistematici).

Figura 6: Descompunerea indicatorului Theil IC pentru modelele comparate


After analyzing the figures, it turns out that the one that best corresponds to the properties of the forecast accuracy indicator (Theil IC) is the VAR 9 multivariate model, but we cannot reject the VAR 4 multivariate model either. On the other hand, the performance of the univariate models ARIMA 1 and ARIMA 2 is disappointing having values of the Theil IC indicator close to 1 (on average 0.8), which shows that these forecast models have a relatively small connection with the historical series. . The VAR 4 model presents fairly satisfactory short-term forecast errors with an average of Theil IC indicator value of 0.2, but in the long term this value increases over 0.4, which shows that the forecasts made on the respective data sample are departs from the historical series of the eur / usd exchange rate. Returning to the VAR 9 model, its performance is given by the inclusion of several variables with significant impact on the exchange rate. From the point of view of the quantity of Theil IC indicator, the VAR 9 model is at acceptable values with an average slightly above 0.2, and from the point of view of the quality of Theil IC indicator, the model is quite good due to the fact that error forecasting is largely explained by non-systematic factors (cov), which it cannot surprise. The remainder of the forecast error is explained to a small extent by the fact that the model does not capture the mean and variance (var) of the real series.

Conclusions From this article, which is based on a series of concrete elements that have been subjected to the study, a series of theoretical and practical conclusions are drawn. The first is that in order to achieve a complex and correct forecast, it is also necessary to carry out a forecast of the exchange rate, because the countries need international cooperation and exchanges. Multivariate models are those that give significance through a matrix of the coefficients calculated in terms of exchange rate perspective. Of course, the exchange rate, known and forecast, does not solve some of the problems encountered in international trade, but it offers the possibility to appreciate in the macroeconomic forecast the effect that the imports and exports made in other fields have. Another conclusion is that a complex forecast cannot but be based on a forecast of the exchange rate, which is quite edifying in terms of knowing the concrete effect that international economic exchanges have. From a practical point of view, the models identified and presented must be based on a careful study of the phenomenon for identifying statistical variables and for constructing models, which resolved to lead to obtaining the regression


coefficients and based on which to identify the trend that the exchange rate will follow in the future.

References 1. Anghelache Constantin, Anghel Mădălina Gabriela (2016). Bazele statisticii

economice. Concepte teoretice şi studii de caz, Editura Economică, Bucureşti, 399 pp

2. Anghelache Constantin, Partachi Ion, Anghel Mădălina-Gabriela (2017). Previzionarea creşterii economice / Forecasting economic growth, Economica, Scientific and didactic journal, Year XXIV, nr. 2 (100), June 2017, pp. 147-152

3. Cheung, Yin-Wong, Menzie D. Chinn, şi Antonio Garcia Pascual (2005), “Empirical exchange rate models of the nineties: Are any fit to survive?”, Journal of International Money and Finance, no.24, 1150-1175

4. Driver, Rebeca L., P. F. Westaway (2004), “Concepts of equilibrium exchange rates”, Bank of England, Working Paper no. 248

5. Gutierrez, Carrasco, Enrique, Carlos, Reinaldo Castro Souza şi Osmani Teixeira de Carvalho Guielen (2007), ’’Selection of Optimal Lag Length in Cointegrated VAR Models with Weak Form of Common Cyclical Features”, Banco Central do Brasil Working Series 139, June 2007

6. Iacob, S,.V. (2019), Utilizarea metodelor statistico-econometrice și econofizice în analize economice, Ed. Economică, 216 pp

7. MacDonald, Ronald (2000), “Concepts to calculate equilibrium exchange rates: an overview”, Deutsche Bundesbank, Discussion Paper no. 3/2000

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