Forecast of Sarima Models: Αn Application to Unemployment Rates of Greece

Chaido Dritsaki

American Journal of Applied Mathematics and Statistics

Forecast of Sarima Models: Αn Application to Unemployment Rates of Greece

Chaido Dritsaki

Department of Accounting and Finance, University of Applied Sciences of Western Macedonia, Kozani, Greece

Abstract

The low unemployment rate is one of the main targets of macroeconomic policy for each government. Forecasting unemployment rate is of great importance for each country so as the government can draw up strategies for fiscal policy. The aim of the paper is to find the most suitable model which is adjusted on unemployment rates of Greece using Box-Jenkins methodology and to examine the precision of forecasting on this model. Models’ estimation was made using the non-linear Maximum likelihood optimization methodology (maximum likelihood–ML), whereas covariance matrix is estimated with OPG method using the numerical optimization of Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. Forecasting unemployment rate was made both with dynamic and static process using all criteria of forecasting measures.

Cite this article:

  • Chaido Dritsaki. Forecast of Sarima Models: Αn Application to Unemployment Rates of Greece. American Journal of Applied Mathematics and Statistics. Vol. 4, No. 5, 2016, pp 136-148. http://pubs.sciepub.com/ajams/4/5/1
  • Dritsaki, Chaido. "Forecast of Sarima Models: Αn Application to Unemployment Rates of Greece." American Journal of Applied Mathematics and Statistics 4.5 (2016): 136-148.
  • Dritsaki, C. (2016). Forecast of Sarima Models: Αn Application to Unemployment Rates of Greece. American Journal of Applied Mathematics and Statistics, 4(5), 136-148.
  • Dritsaki, Chaido. "Forecast of Sarima Models: Αn Application to Unemployment Rates of Greece." American Journal of Applied Mathematics and Statistics 4, no. 5 (2016): 136-148.

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At a glance: Figures

1. Introduction

Every economy has a particular population size. Population is distinguished between economically active and non active for economic reasons. Economically active population is the labor force of economy. Thus, labor force considers those that can and want to work. Labor force is divided in two categories: those who already work and called employed and those who don’t work and are called unemployed. Unemployed are those who can and want to work but cannot find a job.

Unemployment has three basic economic consequences:

• It is regarded as a loss of productive powers

• It means wage loss

• Puts a strain on government’s budget due to unemployment benefits.

The consequences of unemployment are certainly wider because not only it reduces wages but also decreases social position, creates self-respect problems and family matters. In other words, unemployment leads to serious social problems apart from economic ones.

The opposition of unemployment is extremely difficult. The measures taken from various governments are divided in two general categories:

1) The measures taken for the increase of total demand and

2) The measures for professional training and re-education on labor force.

The measures for total demand are fiscal and monetary. Fiscal measures consist of the increase of government expenditure for public works and the promotion for investments. The aim of these works is the direct growth of employment and wages. Monetary measures aim at interest rate reduction in order to strengthen private investments, production and therefore employment. Fiscal and monetary measures aim at the increase of total demand, thus unemployment decrease which is due to insufficient demand. The insufficient demand is the Keynesian unemployment which comes from the fall of economic activity during the recession stage of economic cycle.

Measures of professional formation and re-education facilitate unemployed in the acquisition of professional knowledge and specialization which are necessary in order to be occupied in existing vacancies. It is obvious that these measures aim at the reduction of structural unemployment, created by the disproportion between supply and demand of various specialization. This reduction demands re-education of unemployed in order to acquire the necessary skills where there is scarcity.

The measure of unemployment depends on the size of labor force. So, unemployment is measured as percentage (%) of labor force. The unemployment rate is:

The progress of unemployment rate has been the central issue of political discussion for many developed countries. However, the behavior of unemployment rate during recession and recovery that followed puzzled researchers and policy makers if there is a change on long run trend of unemployment rate. Given this new situation, policy focuses on the dynamics of unemployment rate after the recession so that they can forecast unemployment rate. Forecasting unemployment rate came to the front from policy makers using autoregressive models as well as structural econometric forecasting models.

Greek economy, reached high rates of growth until 2008 but on 2009 there was a downturn, as a result of the international financial crisis whereas from 2010 and afterwards the downturn grew worse due to fiscal imbalances. The need of reform led the country to a mechanism of economic support which consisted of European Union, International Monetary Fund and European Central Bank. Strict income policy and drastic constraints on public expenses, during the last five years, affected negatively GDP’s progress. As a consequence, GDP reduced by 5,4% on 2010, by 8,9% on 2011, by 6,6% on 2012 and by 3,9% on 2013 (constant prices 2010).

Until 2008, unemployment in Greece was relatively low and moved at about 7,8% on average in Eurozone. On 2009, unemployment in Greece increased as a result of international crisis and reached 9,6% while for 2010 it increased further on 12,7% as a result of restrictive fiscal policy that implemented because of debt crisis. During 2011, unemployment rate reached 17,9%, as a consequence of the crisis on Greek economy and the measures taken for fiscal smoothing, while during 2012 exceeded 24% and during 2013 was 27,5%. During 2014 it was noted, for the first time, a slight decrease, even if unemployment remained on high levels about 26,5%. Finally, in September 2015, unemployment reduced and reached 24,6% according to the Hellenic Statistical Authority.

The aim of this paper is to construct the most suitable model in order to investigate and forecast unemployment rates. For this reason the SARIMA models and Box-Jenkins methodology were used, while the forecasting of the models is examined both on dynamic and static process, employing all the criteria forecasting measures. The rest of the paper is organized as follows: the second chapter presents literature review. Following, the Box-Jenkins methodology is provided. On chapter four data and empirical results are presented and on the last chapter the conclusions of the paper are given.

2. Review of Literature

[2] created a family of models known as AutoRegressive Integrated Moving Average (ARIMA) models. These models are applicable to a wide variety of situations. Box and Jenkins have also developed a practical process for the selection of the most suitable ARIMA model out of this family of ARIMA models. Many researchers claim that the creation of an ARIMA model needs judgement and experience.

ARIMA models are suitable for short run forecasts. This is due to the fact that ARIMA models give more emphasis on the recent past rather than distant past. According to [26], long run forecasts on ARIMA models are less reliable than short run. Seasonal AutoRegressive Integrated Moving Average (SARIMA) model is an expansion of simple ARIMA models and contains seasonal and non-seasonal data. SARIMA models have been applied for inflation forecasting ([13, 16, 21]), for exchange rate forecasting ([10, 11]), for tourist arrivals and revenues forecasting ([4, 31]) as well as unemployment forecasting.

Few papers related with unemployment forecasting and Box-Jenkins methodology or ARIMA and SARIMA models have been published. Some of them are the following:

[15] examines the unemployment in Germany using monthly data from January 1965 until November 1989. He uses both ARIMA and VAR models. The comparison of the results for the forecasting of the two examined models shows the advantages and disadvantages of the two methods.

[8] use monthly data for Romania for the period 1998-2007. On their paper, employing Box-Jenkins methodology, they present that the most suitable model is ARIMA (2,1,2). Following with this model they forecast Romania’s unemployment for the following months of 2008.

[12] following Box-Jenkins methodology and using monthly data for the unemployment of Nigeria find that the best model is ARIMA (1,2,1) for the data used. With this model they forecast the unemployment for the following months of Nigeria.

[24] uses quarterly data for the period 1976Q1 – 2011Q4 to examine the forecast of unemployment in Nigeria. Among other models used, he proved that the most suitable for unemployment forecasting in Nigeria is the ARIMA(1,1,2)-ARCH(1) model instead of that of Etuk et al. (2012) that supported on their paper.

[29] on her paper deals with the modeling of employment market on Czech Republic. Box-Jenkins methodology or ARIMA model, are the approaches which uses for modeling time series. Particularly, for unemployment’s model, data from January 2004 until April 2012 are used and with SARIMA model (1,1,0) (1,1,0)12 she forecasts unemployment rate until December 2012.

[18] in order to examine unemployment rate in Thailand they use two techniques: Box-Jenkins and Neural Networks. Their results showed that Box-Jenkins technique proved more efficient for the estimation of unemployment rate in Thailand. Forecasted values that were estimated were consistent with the actual values for unemployment rate.

Finally, [25] using Phillips curve examines unemployment rates and inflation for USA from January 1980 to April 2015. Examining these variables with ARIMA and VAR models, she concluded that VAR models give better forecast than ARIMA.

3. Theoretical Background

[2] and [3] on their papers referred the procedures for the construction of ARIMA models. Seasonal ARIMA models consist of both seasonal and non-seasonal factors in a multiplicative model. ARIMA models, which were first introduced by Box-Jenkins, aimed at time-series forecasting when they became stationary by differencing. A time-series can have seasonal and non-seasonal characteristics. A series has seasonal characteristics when these are repeated over s time periods. Furthermore, in a seasonal series there is often a different mean value between seasonal intervals. Thus, in most cases, seasonal time series are non-stationary.

3.1. Non-seasonal ARIMA Model

A non-seasonal ARIMA model is symbolized as ARIMA (p,d,q) where p is the number of autoregressive lag, d is the differencing lag and q is the moving average lag and can be written as:

(1)

Equation (1) can be also written as:

(2)

where

and

If Zt is a stationary series obtained after d differencing from Yt series then we get:

(3)

so the final form of ARIMA (p,d,q) model can be shaped as:

(4)

The above model is a popular time-series forecasting technique used on a large scale from analysts. In other words, this technique takes into account the historical data and is decomposed in an Autoregressive (AR) process where there is a memory from previous values, an integrated procedure that represents data stationarity and a Moving-Average (MA) procedure which represents the terms of previous error in order to make forecasting easier.

3.2. Seasonal ARIMA Model

A time-series is called seasonal if there is at least one seasonal autoregressive parameter P (SAR) or at least one seasonal moving average parameter Q (SMA) or both parameters (P,Q). Seasonal ARMA (P,Q) is used when seasonal (hence non stationary) behavior is present in the time series. ARMA (P,Q) model can be written as follows:

(5)

where

s=number of periods per season.

Seasonal differencing may be in order if the seasonal component follows a random walk, as in:

(6)

The seasonal difference of order D is defined as:

(7)

So the final form of SARIMA (p,d,q) model X (P,D,Q)s can be formed as:

(8)

where

is an ARMA model with lots of coefficients set to zero.

3.3. Procedure for SARIMA Modeling

• We test diagrammatically the data for the presence of seasonal fluctuations as well as for a possible trend.

• We observe the data correlogram. The coefficients can present slow or quick drop in a expontential or corrugated way.

• If for any lag k=s, the respective coefficient is quite strong in relation to its neighbouring, we consider that the model has seasonality s. Then we isolate correlations coefficients for k=s, 2s, 3s and if they diminish in a slow measure then we get seasonal differences in order to determine the number D of ARIMA seasonal model (P,D,Q)s which is adjusted on data.

• If the existence of trend is obvious, we get the differences on observations till we achieve stationarity. If autocorrelation on first data of the series is strong, then seasonal correlations are becoming obvious on autocorrelation diagrams after differencing or generally on differences order d.

• When we get the required differences (seasonal and non-seasonal), we examine the new autocorrelation and partial autocorrelation diagrams of data on differences for identification on p,q and P,Q order on multiplicative ARIMA model (p,d,q)(P,D,Q)s.

• For facilitation, we can isolate the autocorrelation coefficients with seasonal lags s, 2s, 3s in order to determine the values of P and Q of seasonal ARIMA (P,D,Q)s.

A seasonally SARIMA model is symbolized as SARIMA (P,D,Q) where P is the number of autoregressive lag, D is the differencing lag and Q is the moving average lag and can be written as follows:

(9)
3.4. Estimation of the Model SARIMA

For the estimation of SARIMA models we use the Maximum Likelihood –ML method, where is the estimator of a matrix of parameters and can be approximated by a with mean and covariance matrix and is the following:

(10)

where is the log-likelihood of one observation from the sample, evaluated at the parameter , and is the vector of first derivatives of the log-likelihood.

For the estimation of the asymptotic covariance matrix (10) the Outer Product of Gradients (OPG) estimate is used and is computed as:

(11)

Provided some regularity conditions are satisfied, the OPG estimator is a consistent estimator of . (see [15]).

Also, for the optimization of matrix we use the algorithm of Broyden–Fletcher–Goldfarb–Shanno (BFGS). On numerical optimization, the algorithm BFGS is an for solving unconstrained problems and was developed by [5, 14, 17] and [28].

3.5. Diagnostic Checking of the Model SARIMA

There are several diagnostic tests for the analysis of models. A statistical tool which can be used to determine whether the series present autocorrelation or heteroscedasticity is Q statistic of [19].

(12)

where is the residual autocorrelation at lag , is the number of residuals, is the number of time lags including in the test. The model is considered adequate only if the p value associated with the Ljung-Box Q statistic is higher than a given significance.

The correlogram of the residuals can be used to check residuals’ autocorrelation.

If there is no serial correlation, the autocorrelations and partial autocorrelations at all lags should be nearly zero, and all Q-Statistics should be insignificant with large probability-values.

The correlograms of the squared residuals can be used to check autoregressive conditional heteroskedasticity (ARCH) in the residuals. If there is no ARCH in the residuals, the autocorrelations and partial autocorrelations should be zero at all lags and the Q-statistics should not be significant.

3.6. Forecasting Performance

The forecasting on seasonal ARIMA models is computed for both in sample and out sample values. The optimum forecast value is evaluated from mean squared error (MSE) which measures the average of squared error over the sample period. Other measures (indices) often used for the return of forecasting are the Root Mean Square Error (RMSE), the Mean Absolute Percentage Error (MAPE) and Theil’s inequality index [30].

These indices are taken from the following functions:

(13)
(14)
(15)
(16)

Theil’s inequality index is taken from the following function:

(17)

where

: Actual value of endogenous variable at time t.

: Redacted value of endogenous variable at time t.

T: Number of observations in the simulations (of the sample).

If Theil’s unequal index is U=0, then actual values of the series will be equal with the estimated = for all t, so in this case we can consider that there is a “perfect fit” between actual and predicted data. On the contrary, if coefficient U=1, there is wrong forecasting for the examined model. Afterwards, we present individual Theil’s indices called “unequal ratios” and are the following:

• Bias proportion: indicates the systematic differences in actual and forecasted values.

(18)

where and are the means of the series of and respectively. Bias proportion measures the distance between the mean of simulated series from the mean of actual series.

Variance proportion: indicates unequal variances of actual and forecasted values.

(19)

where and are the standard deviations of the series of and respectively. Variance proportion measures the distance between the variance of simulated series from the variance of actual series.

• Covariance proportion: indicates the correlation between the actual and forecasted values (zero=perfect correlation between actual and forecasted values).

(20)

where ρ is the correlation coefficient between and . Covariance proportion measures the rest of non-systematic error of simulating.

The forecasting ability of a model is satisfying when bias proportions and variance proportions are small. The relationship among the above proportions are UM+US+UC=1 (see [9]).

4. Data and Empirical Results

The variable used in the analysis of the paper is unemployment rate and covers the period from April 1998 until September 2015, total 210 monthly observations. Data derived from OECD database.

4.1. Testing for Non-stationarity

Figure 1 and Figure 2 show Greece’s monthly unemployment rates and the trend analysis respectively. Diagrammatic test is made in order to examine the existence of seasonal fluctuations as well as a possible trend. Following diagram 3 describes the function of autocorrelation and partial autocorrelation respectively.

From Figure 1 we can see that the original data show changeable variance. Also, trend analysis from Figure 2 shows an upward trend. However, on Figure 3, the coefficients on autocorrelation function have a slow fall confirming that the series is non-stationary. Afterwards, we get the first differences of the series and examine stationarity. Figure 4 and Figure 5 present monthly unemployment rates and trend analysis on first differences respectively.

Figure 1. Time series plot of Greece monthly unemployment rate (Linear Trend Model UNEt=5.860+0.078*t)
Figure 2. Trend plot analysis of Greece monthly unemployment rate (Linear Trend Model UNEt=5.860+0.078*t)
Figure 3. Autocorrelation and Partial Correlation Plot of Greece’s monthly unemployment rate
Figure 4. Time series plot of first difference of the original data (Linear Trend Model ΔUNEt=-0.041+0.001*t)

From Figure 4 and Figure 5 we notice that stationarity has not been achieved and seasonality is not obvious. Furthermore, trend analysis on Figure 5 show that there is an upward trend. On Figure 6 we present autocorrelation and partial autocorrelation functions that correspond on the first differences of the series.

Figure 5. Trend analysis of first difference of the original data (Linear Trend Model ΔUNEt=-0.041+0.001*t)
Figure 6. Autocorrelation and partial function of first differences of the original data

From the above figure, the coefficients on autocorrelation function present a slow downturn confirming that the series is not stationary on first differences. Thus, we get second differences.

Figure 7. Time series plot of second differences of the original data (Linear Trend Model Δ2UNEt=0.0009-1.8E-0.5*t)

Figure 7 and Figure 8 show monthly unemployment rates and trend analysis on second differences respectively.

Figure 8. Trend analysis for second differences of the original data (Linear Trend Model Δ2UNEt=0.0009-1.8E-0.5*t)

From Figure 7 and Figure 8 we can see that stationarity has been achieved as there is no trend, and seasonality is obvious. On Figure 9, autocorrelation and partial autocorrelation functions are shown on second differences of series respectively.

Figure 9. Autocorrelation and partial function of second difference of the original data

Coefficients of autocorrelation function show a quick fall on Figure 9, thus series is stationary on second differences. Afterwards, we test for series stationarity using [6, 7] test and [27] unit roots tests.

The results of Augmented Dickey–Fuller (ADF) test and Phillips-Perron (PP) test on unemployment rate series are represented on Table 1.

Table 1. ADF and Phillip-Perron Test on Unemployment Series

The results on Table 1 indicate that unemployment rate is stationary in second differences. Therefore for our model ARIMA (p,d,q) we will have the value d=2.

4.2. Identification of the Model

After the detection of stationarity of the series, we define the form of ARIMA (p,q) models from the correlogram on Figure 9. Parameters p and q can be assessed from partial autocorrelation and autocorrelation coefficients respectively, comparing them with critical value. The limits for both functions (ACF, PACF) are . From the column of autocorrelation in figure 9 we can notice that only the value of the coefficient (autocorrelation coefficient) is greater from the value , while from the column of the coefficients of partial autocorrelation the values (partial autocorrelation coefficients) is greater than the value . Therefore, the value of will be , and respectively, the value of will be . Thereafter we create Table 2 with the values of and as follows:

Table 2. Comparison of models within the range of exploration using AIC, SIC and HQ

The results from Table 2 indicate that according to Akaike (AIC), Schwartz (SIC) and Hannan-Quinn (HQ) criteria, ARΙMA(1,2,1) and ARIMA(0,2,1) models are the most suitable.

4.3. Seasonal Autoregressive Models

Continued on Figure 10 and Figure 11, we present the seasonal difference and the trend analysis on second differences of the series.

Figure 10. Time series plot of the seasonal difference of the second difference data (lag=12)

From the above figures we notice that there is stability both in seasonal and non seasonal level and also the trend is stable (no rise or fall) showing us that there is stationarity on the mean.

Figure 11. Trend analysis of the seasonal difference of the second difference data (Linear trend model D2SUNEt =0.088-0.0015*t)

On Figure 12 the autocorrelation and partial autocorrelation functions of seasonal difference appear and correspond to the second differences of the series.

Figure 12. Autocorrelation and partial function of seasonal difference of the second difference data

From this figure we see that seasonal lags on autocorrelation function is important on lag 11, whereas on partial autocorrelation function is on 11 and 23 lag. This fact denotes that 0<P<2, 0<Q<1.

Thereafter we create Table 3 with the values of and as follows:

Table 3. Comparison of models within the range of exploration using AIC, SIC and HQ

The results from Table 3 indicate that according to the criteria of Akaike (AIC), Schwartz (SIC) and Hannan-Quinn (HQ) the model SARIMA is formulated to SARΙMA (1,2,1) (1,2,1)12 and SARIMA (1,2,1) (0,2,1)12 are the most suitable.

Table 4. Comparison of models within the range of exploration using AIC, SIC and HQ

From Table 4 we notice that according to the criteria of Akaike (AIC), Schwartz (SIC) and Hannan-Quinn (HQ) the model SARIMA (0,2,1) (0,2,1)12 and SARIMA (0,2,1) (1,2,1)12 are the most suitable.

We then proceed to the next stage of the Box-Jenkins approach which is the estimation of the models.

4.4. Estimation of the Model

Thereafter we can proceed to estimating the above model. The following Table 5, Table 6, Table 7 and Table 8 present the results of these models.

Table 5. Estimation Model SARIMA(1,2,1)(1,2,1)12

Results on Table 5 show that the coefficient on AR(1) parameter is not statistically significant.

Table 6. Estimation Model SARIMA(1,2,1)(0,2,1)12

Results on Table 6 show that the coefficient on AR(1) and SMA(12) parameters are not statistically significant on 5% level of significance.

Table 7. Estimation Model SARIMA(0,2,1)(0,2,1)12

Results on Table 7 show that coefficient on SMA(12) parameter is not statistically significant on 5% level of significance.

Table 8. Estimation Model SARIMA(0,2,1)(1,2,1)12

The results of the above table show that all coefficients are statistically significant thus that model can be used for forecasting.

4.5. Diagnostic Checking of the Model SARIMA(0,2,1)(1,2,1)12

On the Figure 13 and Figure 14, the residuals test for the autocorrelation with conditional heteroscedasticity (ARCH model) is provided.

From the results of Figure 13 and Figure 14, we can see that autocorrelation and partial autocorrelation coefficients are non statistical significant in all lags as all Q-statistics have large probability-values. So, we can regard that residuals are not autocorrelated and don’t form ARCH models. Thus, SARIMA (0,2,1)(1,2,1)12 model can be used for forecasting.

4.6. Forecasting.

For forecasting SARIMA(0,2,1)(1,2,1)12 model we use both dynamic and static forecasting procedure. The dynamic procedure calculates forecasts for periods after the first period in the sample by using the previously forecasted values from lagged dependent variable and ARMA terms. This procedure is called n-step ahead forecast. Static procedure uses actual and non forecasted values of dependent variable. This procedure is called one step- ahead forecast.

In Figure 15 and Figure 16 we represent the criteria for the evaluation of the forecasts in and out sample forecast at level form of the dependent variable, using dynamic and static forecast respectively.

Figure 13. Diagnostic residuals’ autocorrelation test of SARIMA(0,2,1)(1,2,1)12 model
Figure 14. Diagnostic test for residuals’ conditional autocorrelation of SARIMA(0,2,1)(1,2,1)12 model

From Figure 15 and Figure 16 we notice that the indices of root mean squared error and mean absolute error get smaller values on Figure 16 than that of Figure 15. We conclude that static procedure gives better forecasting ability on the examined model. Furthermore, Theil’s Inequality index on the static procedure is smaller compared to the dynamic procedure and is close to zero. This indication on Theil’s index shows the “goodness of fit” for the model. Moreover, the proportion of bias (it measures how far is the mean of simulated series from the mean of the actual series), and the proportion of variance (it measures how far is the variance of simulated series from the variance of actual series), has very small value on the static procedure. These indices prove that the forecasting ability of static procedure show more accurate forecasting. On the contrary, the proportion of covariance which measures the rest of non-systematic forecast error, is larger on the static procedure as it was expected.

Table 9 displays the comparison between the actual, forecasted and the residual starting from 2013:10 to 2015:09.

Table 9. Comparison between the actual, forecast values from 2013:10 to 2015:09

From Table 9 we can see that SARIMA(0,2,1)(1,2,1)12 model has the best predictive power. The forecasted value of unemployment, deriving from the suggested model, refers to September 2015 and is 24.62%. This value is very close to the actual which is 24.6%. Therefore, SARIMA(0,2,1)(1,2,1)12 model that we suggest has a good and precise forecasting for unemployment in Greece.

5. Conclusion

Unemployment plagues many countries so it is important to capture the trend of this series. The use of ARIMA models is a highly flexible tool in order to forecast unemployment rate if there is no government’s intervention which will change this trend. The main goal of this paper is to find the most suitable model with a forecasting ability in order to forecast unemployment in Greece. Using Box-Jenkins methodology, we determined the form of SARIMA model and estimated this model with the non-linear optimization method of Maximum Likelihood, using numerical optimization Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. For the forecasting power of the model, both the dynamic and static procedure together with the criteria forecasting measures, were used. The results of the forecast showed that the forecasted value of unemployment is close to the actual value. This result showed that model’s suitability can be used to forecast unemployment in Greece for the following years.

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