# A comparative analysis of alternative univariate time series models in forecasting Turkish inflation.

1. Introduction

The stability of prices, as an important indicator of overall economic performance is one of the main objectives of monetary policy. Turkish economy has a long history of instable macroeconomic performance, especially the persistence of high inflation rates, which lasted for almost five decades. During this long period, many stability programs, mostly backed by the IMF, could not be implemented thoroughly, and failed to achieve the desired outcome, economic and price stability. After the collapse of IMF-supported program in February 2001, Turkey faced a financial turmoil, followed by a destructive overall economic crisis. Consuming all other options, Central Bank of The Republic of Turkey (CBRT) announced to implement new policies that focus on monetary targeting and at the same time on the future inflation. The meaning of this policy was an "implicit" inflation targeting, and in 2005 the CBRT started to implement "explicit" inflation targeting policy, assuming that the conditions of success were available.

Although, inflation targeting was started on a trial and error basis of the central banks in New Zealand, Canada, England and Sweden with little academic and theoretical background, its popularity has let a growing academic effort to analyze the effectiveness and overall macroeconomic effects of these policies. The main characteristics of inflation targeting policy are described by Mishkin (2000) as; the public announcement of a well defined numerical target for inflation; a commitment to price stability as the primary goal of monetary policy; an information inclusive strategy for deciding the setting of policy instruments; increased transparency of the monetary policy strategy; and increased accountability of the central bank for attaining its inflation objectives.

While predicting future values of price changes accurately is essential for inflation targeting policy, and also for the credibility of monetary policy, there is no consensus on a superior methodology. The reason is that the behavior of price movements is complex, volatile and nonlinear in a wide range. The empirical literature on forecasting inflation rates suggest that there is no single methodology that can outperform others in a case independent context. The recent advance in the applicability of nonlinear models has changed the path of the discussion mainly on the comparison of linear and nonlinear models to describe the behavior of time series. Among others, artificial neural network models (ANNs) have emerged as a powerful statistical modeling technique for nonlinear modeling of time series and applied into various case studies (Gonzalez 2000; Tkacz 2001; Zhang et al. 1998). Although these models have numerous advantages such as letting more general and flexible formations that can approximate any continuous measurable function with arbitrarily desired accuracy level (Cybenko 1989; Hornik et al. 1989; Hornik 1993; Koutroumanidisa et al. 2011), the theoretical superiority of ANNs to the linear models is not yet to be justified in forecasting performances (Moshiri, Cameron 2000; Clements et al. 2004) (1).

Even though a long historical experience with high inflation levels in Turkey, the topic seems to be neglected given its impact on social and economic structure until recently. Domac (2004) estimates various univariate and multivariate linear time series models to understand dynamics of inflation in Turkey using monthly data covering the period 1990-01-2002-12. In this study he estimates mark-up models, monetary models, and the Phillips curve along with the simple univariate autoregressive model. He finds that the mark up models have the best in-sample performance, whereas the Phillips curve and the money gap models turn out to be better in terms of out-of-sample forecasting accuracy. Onder (2004) compares the forecasting performance of the Phillips curve with autoregressive intregrated moving average (ARIMA), vector autoregression, vector error correction and a random walk model in Turkey for the period between 1987 and 2001. She finds that Phillips curve provides better forecasting accuracy than the other alternative models.

In this study we evaluate inflation forecasting accuracy of alternative univariate time series models, i.e. random walk, ARIMA (Autoregressive Integrated Moving Average), ARFIMA (Autoregressive Fractionally Integrated Moving Average), FIGARCH (Fractionally Integrated GARCH), unobserved components models (UCM) and ANNs. We find that when the static forecasts are taken into consideration, ARFIMA and ARIMA models produce better forecasts than UCM and ANN models at earlier forecast horizons. However, in terms of dynamic forecasts UCM and ANN models turn out to have better forecasting accuracy than the other models.

The rest of the paper is organized as follows. First, the basic descriptive properties of the inflation data are presented in Section 2, followed by a brief description of the methodologies used in the study in Section 3. Empirical findings of the study are presented in Section 4. Finally some conclusions and policy proposals are put forward in the last section.

2. Data

This paper uses monthly seasonally adjusted Turkish CPI (Consumer Price Indices) data collected from International Financial Statistics (IFS) online database covering the period 1982:01-2009:12 (2). Inflation ([[pi].sub.t]) is defined as month on month changes in the natural log of CPI (see Figure 1).

Before proceeding, we summarize distribution and time series properties of the inflation (see Table A1 and Figure A1). Skewness and kurtosis coefficients of inflation are higher than that of the standard normal distribution and Jarque-Bera statistic, which has the critical value of 5.99 at 5% significance level, indicates that [[pi].sub.t] is not normally distributed. The same descriptive statistics computed for the period after 2003 suggest that inflation follows relatively stable pattern after the implementation of inflation targeting strategy. Density function and Quantile-Quantile (QQ) plot are further supportive of this fact. ADF unit root test implies the stationarity of inflation. Partial autocorrelation function indicates existence of positive autocorrelation especially at first, fourth, fifth and twelfth lags. The estimated spectral density function which is defined as a smoothed function of the sample autocorrelations shows that [[pi].sub.t] is stationary, but autocorrelated.

[FIGURE 1 OMITTED]

3. Forecasting models

3.1. Artificial neural network models

In recent years artificial neural network models have become a popular tool in the forecasts of financial and macroeconomic time series (3). ANNs are nonlinear models that can be used to explain relations which may be difficult to model with linear constant parameter models (Gonzalez 2000). It contains interconnected group of artificial neurons just as the structure of the human brain that helps to learn the pattern of the data.

In contrast to conventional model-based methodologies, ANNs do not require a priori assumptions about the relationship. Due to its nonlinear structure, they can be approximated into any continuous functions and they are known as universal approximation (Hornik et al. 1989; Hill et al. 1994). In spite of these advantages, forecasting with ANNs is not an easy task since there is no consensus about determining the parameters of the models, such as architecture, size of training and validation sets, and the type of activation function. In most studies those factors, playing vital role in the success of ANNs, are rather determined through successive experiments (Zhang et al. 1998; Panda, Narasimhan 2007).

As far as their architecture is concerned, ANNs can be divided into two categories: Feedback (recurrent) and feed-forward neural networks. Feed-forward ANNs work only one direction from inputs to output, whereas feedback ANNs are also able to move into both directions. Feedback ANNs are not suitable for time series forecasting, since they require twice times higher number of parameters than the feed-forward ANNs. Hence we use fully connected feed-forward network with one hidden layer which is the most popular ANN models for time series forecasting applications (Tkacz 2001; Nakamura 2005). ANN model used in this study can be formulated as (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

where [[delta].sub.0] is a bias vector {[[delta].sub.k], k = 0,1, ..., r} is a vector of weights from the hidden to output nodes and {[[phi].sub.ik], i = 0,1, .., p; k = 1,2, ..., l} are weights from the input to hidden nodes , in our case they are parameters of current and lagged values of inflation. p is the number of input nodes, r is the number of hidden nodes, f is a nonlinear activation function by which information transmitted to the next neuron. Given the availability of various activation functions, such as linear, hyperbolic tangent and logistic, we used the most preferred the logistic sigmoidal function f (x) = 1/1 + [[e.sup.-[lambda]], following the previous literature (Tkacz 2001; Panda, Narasimhan 2007; Binner et al. 2004). In this framework ANN model in (1) can be conceived as a nonlinear autoregressive (AR) model.

3.2. Linear time series models

In order to evaluate forecasting performance of ANNs, we consider three univariate time series models. The first model is the following first order random walk (RW) without drift model of inflation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

where [[rho].sub.1] is autoregressive parameter to be estimated and [u.sub.t] is i.i.d. error term. By adding a constant and moving average term to RW model, Equation (2) can be converted into ARIMA (p, d, q) model as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

Where, p, d and q are integers referring the order of the autoregressive, integrated and moving average parts respectively. Time series properties of inflation in the previous section suggest that inflation data may also exhibit long-memory properties: autocorrelation and spectral density functions decay slowly but there is no indication for non-stationarity. Therefore, in addition to ARMA (p, d, q) model above we also use the long-memory ARFIMA model in forecasting inflation. The ARFIMA (p, d, q) model for [[pi].sub.t] is written as,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

Where [PHI](L) and [THETA](L) refer to autoregressive and moving average polynomials respectively. [(1-L).sup.d] is the fractional difference parameter. The stationarity of [[pi].sub.t] depends on the value of d. ARFIMA(p,d,q) process is said to be covariance stationary if d < 0.5. In that case autocovariance function decays hyperbolically. The process is long memory stationary if 0 < d < 0.5. For -0.5 < d < 0 the process is known as intermediate memory, and when d [greater than or equal to] 0.5 the process is said to be non-stationary.

The descriptive analysis of [[pi].sub.t] also implies that the distribution of inflation shows non-gaussian behavior, it is right skewed and has fat tails. In order to account for this type of asymmetry, one solution is modeling [[pi].sub.t] with GARCH-type models with a skewed density function. For this purpose, we employed Fractionally Integrated GARCH (FIGARCH) model introduced by Baillie et al. (1996). This model allows us to analyze the dynamic adjustments of both the conditional mean and the conditional variance of inflation over time. Following Lambert and Laurent (2000) non-normality of inflation is considered by employing skewed-student density function instead of normal density in the maximum likelihood estimates.

The mean equation of the FIGARCH is identical to ARIMA specification in (3). Additionally, the conditional variance equation of the model is formulated as follows,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

where [d.sub.v] is a fractional parameter of the variance equation ranging from 0 to 1 and [alpha] and [beta] are the respective ARCH and GARCH parameters satisfying [alpha], [beta] < 1 condition.

The functional form of FIGARCH model presented in (3) and (5) is very flexible (5). For [d.sub.v] = 0 the specification collapses into conventional GARCH model, whereas for [d.sub.v] = 1 model becomes an Integrated GARCH. The process is reduces to ARFIMA model when [h.sub.t] a constant.

In addition to autoregressive time series models, we also consider a univariate unobserved components model (UCM) of inflation, also known as local level model, having the following state space form (Harvey 1989; Commandeur, Koopman 2007),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

Where [[mu].sub.t] is time varying unobserved level, [[epsilon].sub.t] and [[eta].sub.t] are i.i.d. disturbances of measurement and transition equations. This model allows us to decompose inflation in two parts as level mt long run component and irregular components [[epsilon].sub.t] and [[eta].sub.t]. The second equation in (6) known as transition equation indicates that unobserved level of inflation is time varying and assumed to follow first order random walk process over time. The ratio between variances of the level and irregular components [[sigma].sup.2.sub.[eta]]/[[sigma].sup.2.sub.[epsilon]] is called signal to noise ratio. After maximum likelihood estimation of the parameter level [[mu].sub.t] is obtained through a Kalman filter. This process is also known as smoothing or signal extraction.

4. Empirical results

4.1. Model estimations

In this section, the models defined in the previous section are estimated. First we start from the neural network model. The estimation of the ANNs includes two steps. Training stage involves an iterative procedure where optimum hidden node and output node weights are adjusted by means of neural network algorithm. After the training, the model can be used to produce fitted or forecasted values. The sample period up to 2007:12 is accepted as the training set for ANN model, the remaining 24 observations are reserved to testing set.

We use backpropagation techniques with an adaptive learning rate algorithm to train the model to specified level of convergence. As a standard procedure, [[pi].sub.t] is mapped linearly into the range 0 and 1 to adjust for the logistic activation function (6). Our model is trained until the mean square error, the mean squared error between the actual and the fitted values of the output of the ANN, lies below the convergence criterion. As previously stated, determining the number of input and output nodes of the network is very important to produce good forecasts. We conduct estimations to find out whether a change in the number of input and output nodes makes any effect on the training performance. For this purpose RMSE values obtained from the training set of the models that contain different combinations of input and hidden nodes are reported in Table 1. RMSE values decrease with the increase in the number of input and hidden nodes used in the estimation. RMSE is minimized when the model is estimated with 10 and 11 input nodes. Hence we retain the ANN model with 11 and 4 hidden nodes as a best performer in training to use for the out of sample forecast comparison.

After the ANN, we estimate RW and ARIMA models. We do not employ Box and Jenkins (1976) approach in the modeling. From the forecasting point of view it may be disadvantageous to select p and q arbitrarily large. A higher order model will produce lower error give the best fit in sample, but when the model is used for out of sample forecasting purpose, it is likely to produce worse forecast than the lower order model, since the mean square error of the forecasts errors will not affected by only the stationary variance of the model but also by errors arising from the estimation of the parameters of the model (Brockwell, Davis 2002). Therefore we estimate different combinations of ARIMA models with maximum lag order for p = 15 and q = 2 to overcome this problem. ARIMA (8, 0, 0) is found to be the best model selected by Akaike Information Criterion (See Table 2)7. The sum of autoregressive parameters is obtained as 0.834 confirming the persistent behavior of inflation over the investigation period. Diagnostic tests reported in Table 2 show that the model suffers from non-normality. They also present some evidence for the first order ARCH effect, indicating that the variance of the error term is not constant over time. However we do not find any sign of autocorrelation in the residuals.

The maximum likelihood estimation of the ARFIMA is shown in Table 3. As compared to RW and ARIMA models persistence of inflation remains low, however the model provides similar fit to the data in terms of log-likelihood and Akaike information criteria. The estimated long memory conditional mean parameter d is 0.334 and significantly different from zero. Since the parameter lies in the range 0 < d < 0.5, one can say that inflation in Turkey follows a long memory stationary process.

We also estimate FIGARCH model explained above with maximum likelihood method based on Skewed Student distribution, with 2.9092 degrees of freedom. On general, the parameter estimates of the FIGARCH are found to be significant and corroborate the existence of long memory effects as in the ARFIMA model. The inclusion of conditional heteroscedasticity in the residuals improves fit of the model as evidenced by AIC and log-likelihood values (see Table 4). The model also yields a very high and significant fractional difference parameter [d.sub.v] with 0.775, suggesting that conditional variance of the inflation is characterized by a near integrated GARCH process. The estimated parameter of the skewed student distribution [psi] known as the asymmetry parameter within the range of -1 < [psi] < 1 is found to be positive and significant supporting right skewed characteristics of the data. Another distribution parameter of the distribution v also confirms heavy-tailed characteristics of the inflation (8).

Finally we estimate unobserved components model of inflation based on (6). The results of the estimated state vector by maximum likelihood are presented in Table 5. Local level model of inflation passed all diagnostic tests except for normality of the residuals. However cumulative sum of squared residuals (CUSUM) suggest that the model is stable (see Figure A3). The parameter of the unobserved level of inflation is found to be significant, suggesting that underlying level of inflation follows time varying pattern that can be modeled through the first order random walk process (see Figure 2). Estimated state vector shows that inflation has an increasing trend with about 0.94% on monthly basis.

[FIGURE 2 OMITTED]

4.2. Evaluating forecasting accuracy

The forecasting accuracy of alternative models is compared in terms of both static and dynamic forecasts. A Static (one-step-ahead) forecast uses the actual values for lagged dependent variables to compute forecasted values of dependent variable. Therefore this technique is more convenient for short term forecasting purpose. On the other hand, a dynamic forecast (multi-step ahead) has ability to produce long term projected values in which forecasted values are generated through an iterative procedure, i.e. forecasted values computed in the earlier periods are used as independent variables (Diebold 2007; Makridakis et al. 1998). Out of sample forecasted values of inflation are computed for each of the models in static and dynamic context. Besides, to account for the impact of change in forecasting path on the accuracy of alternative models we consider different starting points for each forecasting horizon (See Table 6). Hence we allowed each forecasting to end. All estimations are conducted using the sample period up to 2007:12 (training set for ANN model), the remaining 24 observations are reserved to compare out of sample forecasting accuracy (testing set for ANN model).

In order to evaluate the forecasting accuracy of the models, we consider several evaluation criterions. First we compute Mean Absolute Error (MAE) and Root-Mean Squared Error (RMSE) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

where [[??].sub.t] and [[pi].sub.t] represents forecasted and actual values of inflation respectively; h denotes forecast horizon h = 6, 12, 18, 24. In addition, scale invariant Theil inequality statistics (Theil) is also computed to evaluate forecasting performance (Pindyck, Rubinfeld 1998).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

Theil statictics is always between zero and one, where zero corresponds to perfect fit. Theil inequality statistics can be decomposed into three parts by following two steps. First, mean squared forecast error is divided into three parts:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] denote the means and standard deviations of [[??].sub.t] and [??], and r represents the correlation between actual and forecasted values. Dividing each part of (8) by the mean squared error itself results in the following decomposition of the Theil inequality statistics,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

The first term is the bias proportion, which shows contribution of the difference in means between the forecasted and actual series to the Theil inequality. The second term is known as the variance proportion, it measures the difference in variance between actual and forecasted values. The last term, the covariance proportion shows remaining unsystematic errors in the forecast. Forecasts are considered as good when the bias proportions are found relatively small compared to covariance proportions. Theil inequality statistics should be mostly explained by the covariance proportions (Pindyck, Rubinfeld 1998).

The criterions described above are computed with different horizons to evaluate out-of-sample forecasting accuracy of the alternative models. The findings show that linear models have the lowest RMSE values at shorter forecast horizon, however as forecast horizon gets longer ANN and UCM performs better. Forecasting accuracy statistics are illustrated in Table 6. Random walk without drift model is the worst performer in terms of both static and dynamic forecasts, since it presents highest value of RMSE, MAE and Theil-Inequality statistics at each forecasting horizons. When the one-step ahead forecasts are considered, ARFIMA model turns out be the best performer according to MAE and RMSE criterions (see Figure 3). ARIMA and FIGARCH models are also provide better fit than UCM and ANN at six months. However the performance of UCM has improved considerably with the increase in forecasting horizons, it becomes the second best model after the ARFIMA for the horizons longer than six months.

[FIGURE 3 OMITTED]

The superiority of UCM is more pronounced in terms of dynamic forecasts. The statistics of ANN is very close to that of UCM model up to 18 months. Hence ANN turns out to be the second best model in terms of tracking inflation. In autoregressive models the bias proportion accounts for more than 50 percentage of Theil inequality indicating the presence of systematic error between actual and forecasted values. This also corroborates the evidence on the lower quality of forecasts produced by the other statistics. Overall, the UCM provides superior long-term forecasts compared to the other techniques.

5. Conclusions

The stability of prices, as an important indicator of overall economic performance is one of the main objectives of monetary policy. In recent years, inflation targeting, has been implemented by most of the central banks around the world, as a policy tool of maintaining price stability. However, maintaining price stability over a period requires forward-looking approaches due to the lags and dynamic structure of price movements. Therefore, predicting and forecasting inflation has emerged as a crucial factor to be placed in monetary policy decisions for almost every monetary decision making process. In this paper we evaluate inflation forecasting accuracy of alternative time series models. For this purpose, static and dynamic forecasts are produced and the accuracies are compared by RMSE, MAE and Theil inequality measures. Static forecast uses the actual values in forecasting, and likely to produce more accurate results for short term forecasting. However, a dynamic forecast employs iterated forecast values and more convenient for long term projections.

We find that at earlier forecast horizons conventional autoregressive models, especially ARIMA and ARFIMA, provide better forecasting performance. However when the horizon gets longer, UCM turns out to be the best performer. The accuracy of ANN model has improved considerably with an increase in the forecast horizon. The superiority of UCM over the alternative models suggests that inflation in Turkey has a time varying pattern and linear models fail to track underlying level of inflation.

These findings indicate that the superiority of the models depends on the forecasting horizon. Inflation targeting policies require long term predictability of price changes, policy makers can consider UCM and ANN models as serious candidates that are able to better track the future values of inflation than the conventional autoregressive models.

doi:10.3846/16111699.2011.620135

APPENDIX

[FIGURE A1 OMITTED]

[FIGURE A2 OMITTED]

[FIGURE A3 OMITTED]

Received 26 January 2011; accepted 28 April 2011

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(1) Some studies find evidence in favour of neural network models (see for example Binner et al. 2004; Nakamura 2005; Panda, Narasimhan 2007) whereas the others not (Faraway, Chatfield 1998; Stock, Watson 1998; Heravi et al. 2004).

(2) Series available on IFS database is not originally seasonally adjusted. In order to obtain adjusted CPI series we employ Tramo-Seats method.

(3) See Zhang et al. (1998) for detailed literature survey on forecasting with ANN.

(4) See Figure A2 in Appendix for the general architecture of the feed-forward neural network model.

(5) For detailed information on the estimation of FIGARCH model see Chung (1999).

(6) In order to obtain the data on normalized inflation [[pi].sup.norm.sub.t], we use the following conventional formula (Mehrotra et al. 1996; Panda, Narasimhan 2007) [[pi].sup.norm.sub.t] = ([[pi].sub.t] - [[pi].sup.min.sub.t]) /([[pi].sup.max].sub.t] - [[pi].sup.min.sub.t]) * [[pi].sup.min.sub.t] and [[pi].sup.max.sub.t] denote minimum and maximum values of inflation respectively.

(7) An automated ARIMA model selection procedure written in RATS is used to obtain AIC for all combinations of ARIMA models.

(8) For detailed information on the functional form of skewed student distribution see Lambert and Laurent (2000).

A. Nazif CATIK is an Assistant Professor at Ege University, Turkey, with Ph.D. His scientific interests are related to the nonlinear analysis of macro data and time series. He has written in cited and indexed journals like Emerging Markets Finance and Trade, Applied Economics Letters, The Journal of Developing Areas, Economics Bulletin, while he has been reviewer in various journals such as METU Development Studies, Journal of Economic Studies. He has published papers in national and international conferences on time series analysis and robust statistics.

Mehmet KARACUKA is an Assistant Professor at Ege University, Turkey, with Ph.D. in Economics, and he is teaching Microeconomics and Applied Econometrics. His research interests are Applied Econometric Methods and Industrial Organization. He has written in cited journals like Telecommunications Policy, Applied Economics Letters, The Journal of Developing Areas; while he has been reviewer in different national and international journals such as The Journal of Developing Areas, The Service Industries Journal.

A. Nazif Catik [1], Mehmet Karacuka [2]

Department of Economics, Ege University, 35040 Bornova / Izmir, Turkey E-mails: [1] a.nazif.catik@ege.edu.tr (corresponding author); [2] mehmet.karacuka@ege.edu.tr

The stability of prices, as an important indicator of overall economic performance is one of the main objectives of monetary policy. Turkish economy has a long history of instable macroeconomic performance, especially the persistence of high inflation rates, which lasted for almost five decades. During this long period, many stability programs, mostly backed by the IMF, could not be implemented thoroughly, and failed to achieve the desired outcome, economic and price stability. After the collapse of IMF-supported program in February 2001, Turkey faced a financial turmoil, followed by a destructive overall economic crisis. Consuming all other options, Central Bank of The Republic of Turkey (CBRT) announced to implement new policies that focus on monetary targeting and at the same time on the future inflation. The meaning of this policy was an "implicit" inflation targeting, and in 2005 the CBRT started to implement "explicit" inflation targeting policy, assuming that the conditions of success were available.

Although, inflation targeting was started on a trial and error basis of the central banks in New Zealand, Canada, England and Sweden with little academic and theoretical background, its popularity has let a growing academic effort to analyze the effectiveness and overall macroeconomic effects of these policies. The main characteristics of inflation targeting policy are described by Mishkin (2000) as; the public announcement of a well defined numerical target for inflation; a commitment to price stability as the primary goal of monetary policy; an information inclusive strategy for deciding the setting of policy instruments; increased transparency of the monetary policy strategy; and increased accountability of the central bank for attaining its inflation objectives.

While predicting future values of price changes accurately is essential for inflation targeting policy, and also for the credibility of monetary policy, there is no consensus on a superior methodology. The reason is that the behavior of price movements is complex, volatile and nonlinear in a wide range. The empirical literature on forecasting inflation rates suggest that there is no single methodology that can outperform others in a case independent context. The recent advance in the applicability of nonlinear models has changed the path of the discussion mainly on the comparison of linear and nonlinear models to describe the behavior of time series. Among others, artificial neural network models (ANNs) have emerged as a powerful statistical modeling technique for nonlinear modeling of time series and applied into various case studies (Gonzalez 2000; Tkacz 2001; Zhang et al. 1998). Although these models have numerous advantages such as letting more general and flexible formations that can approximate any continuous measurable function with arbitrarily desired accuracy level (Cybenko 1989; Hornik et al. 1989; Hornik 1993; Koutroumanidisa et al. 2011), the theoretical superiority of ANNs to the linear models is not yet to be justified in forecasting performances (Moshiri, Cameron 2000; Clements et al. 2004) (1).

Even though a long historical experience with high inflation levels in Turkey, the topic seems to be neglected given its impact on social and economic structure until recently. Domac (2004) estimates various univariate and multivariate linear time series models to understand dynamics of inflation in Turkey using monthly data covering the period 1990-01-2002-12. In this study he estimates mark-up models, monetary models, and the Phillips curve along with the simple univariate autoregressive model. He finds that the mark up models have the best in-sample performance, whereas the Phillips curve and the money gap models turn out to be better in terms of out-of-sample forecasting accuracy. Onder (2004) compares the forecasting performance of the Phillips curve with autoregressive intregrated moving average (ARIMA), vector autoregression, vector error correction and a random walk model in Turkey for the period between 1987 and 2001. She finds that Phillips curve provides better forecasting accuracy than the other alternative models.

In this study we evaluate inflation forecasting accuracy of alternative univariate time series models, i.e. random walk, ARIMA (Autoregressive Integrated Moving Average), ARFIMA (Autoregressive Fractionally Integrated Moving Average), FIGARCH (Fractionally Integrated GARCH), unobserved components models (UCM) and ANNs. We find that when the static forecasts are taken into consideration, ARFIMA and ARIMA models produce better forecasts than UCM and ANN models at earlier forecast horizons. However, in terms of dynamic forecasts UCM and ANN models turn out to have better forecasting accuracy than the other models.

The rest of the paper is organized as follows. First, the basic descriptive properties of the inflation data are presented in Section 2, followed by a brief description of the methodologies used in the study in Section 3. Empirical findings of the study are presented in Section 4. Finally some conclusions and policy proposals are put forward in the last section.

2. Data

This paper uses monthly seasonally adjusted Turkish CPI (Consumer Price Indices) data collected from International Financial Statistics (IFS) online database covering the period 1982:01-2009:12 (2). Inflation ([[pi].sub.t]) is defined as month on month changes in the natural log of CPI (see Figure 1).

Before proceeding, we summarize distribution and time series properties of the inflation (see Table A1 and Figure A1). Skewness and kurtosis coefficients of inflation are higher than that of the standard normal distribution and Jarque-Bera statistic, which has the critical value of 5.99 at 5% significance level, indicates that [[pi].sub.t] is not normally distributed. The same descriptive statistics computed for the period after 2003 suggest that inflation follows relatively stable pattern after the implementation of inflation targeting strategy. Density function and Quantile-Quantile (QQ) plot are further supportive of this fact. ADF unit root test implies the stationarity of inflation. Partial autocorrelation function indicates existence of positive autocorrelation especially at first, fourth, fifth and twelfth lags. The estimated spectral density function which is defined as a smoothed function of the sample autocorrelations shows that [[pi].sub.t] is stationary, but autocorrelated.

[FIGURE 1 OMITTED]

3. Forecasting models

3.1. Artificial neural network models

In recent years artificial neural network models have become a popular tool in the forecasts of financial and macroeconomic time series (3). ANNs are nonlinear models that can be used to explain relations which may be difficult to model with linear constant parameter models (Gonzalez 2000). It contains interconnected group of artificial neurons just as the structure of the human brain that helps to learn the pattern of the data.

In contrast to conventional model-based methodologies, ANNs do not require a priori assumptions about the relationship. Due to its nonlinear structure, they can be approximated into any continuous functions and they are known as universal approximation (Hornik et al. 1989; Hill et al. 1994). In spite of these advantages, forecasting with ANNs is not an easy task since there is no consensus about determining the parameters of the models, such as architecture, size of training and validation sets, and the type of activation function. In most studies those factors, playing vital role in the success of ANNs, are rather determined through successive experiments (Zhang et al. 1998; Panda, Narasimhan 2007).

As far as their architecture is concerned, ANNs can be divided into two categories: Feedback (recurrent) and feed-forward neural networks. Feed-forward ANNs work only one direction from inputs to output, whereas feedback ANNs are also able to move into both directions. Feedback ANNs are not suitable for time series forecasting, since they require twice times higher number of parameters than the feed-forward ANNs. Hence we use fully connected feed-forward network with one hidden layer which is the most popular ANN models for time series forecasting applications (Tkacz 2001; Nakamura 2005). ANN model used in this study can be formulated as (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

where [[delta].sub.0] is a bias vector {[[delta].sub.k], k = 0,1, ..., r} is a vector of weights from the hidden to output nodes and {[[phi].sub.ik], i = 0,1, .., p; k = 1,2, ..., l} are weights from the input to hidden nodes , in our case they are parameters of current and lagged values of inflation. p is the number of input nodes, r is the number of hidden nodes, f is a nonlinear activation function by which information transmitted to the next neuron. Given the availability of various activation functions, such as linear, hyperbolic tangent and logistic, we used the most preferred the logistic sigmoidal function f (x) = 1/1 + [[e.sup.-[lambda]], following the previous literature (Tkacz 2001; Panda, Narasimhan 2007; Binner et al. 2004). In this framework ANN model in (1) can be conceived as a nonlinear autoregressive (AR) model.

3.2. Linear time series models

In order to evaluate forecasting performance of ANNs, we consider three univariate time series models. The first model is the following first order random walk (RW) without drift model of inflation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

where [[rho].sub.1] is autoregressive parameter to be estimated and [u.sub.t] is i.i.d. error term. By adding a constant and moving average term to RW model, Equation (2) can be converted into ARIMA (p, d, q) model as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

Where, p, d and q are integers referring the order of the autoregressive, integrated and moving average parts respectively. Time series properties of inflation in the previous section suggest that inflation data may also exhibit long-memory properties: autocorrelation and spectral density functions decay slowly but there is no indication for non-stationarity. Therefore, in addition to ARMA (p, d, q) model above we also use the long-memory ARFIMA model in forecasting inflation. The ARFIMA (p, d, q) model for [[pi].sub.t] is written as,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

Where [PHI](L) and [THETA](L) refer to autoregressive and moving average polynomials respectively. [(1-L).sup.d] is the fractional difference parameter. The stationarity of [[pi].sub.t] depends on the value of d. ARFIMA(p,d,q) process is said to be covariance stationary if d < 0.5. In that case autocovariance function decays hyperbolically. The process is long memory stationary if 0 < d < 0.5. For -0.5 < d < 0 the process is known as intermediate memory, and when d [greater than or equal to] 0.5 the process is said to be non-stationary.

The descriptive analysis of [[pi].sub.t] also implies that the distribution of inflation shows non-gaussian behavior, it is right skewed and has fat tails. In order to account for this type of asymmetry, one solution is modeling [[pi].sub.t] with GARCH-type models with a skewed density function. For this purpose, we employed Fractionally Integrated GARCH (FIGARCH) model introduced by Baillie et al. (1996). This model allows us to analyze the dynamic adjustments of both the conditional mean and the conditional variance of inflation over time. Following Lambert and Laurent (2000) non-normality of inflation is considered by employing skewed-student density function instead of normal density in the maximum likelihood estimates.

The mean equation of the FIGARCH is identical to ARIMA specification in (3). Additionally, the conditional variance equation of the model is formulated as follows,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

where [d.sub.v] is a fractional parameter of the variance equation ranging from 0 to 1 and [alpha] and [beta] are the respective ARCH and GARCH parameters satisfying [alpha], [beta] < 1 condition.

The functional form of FIGARCH model presented in (3) and (5) is very flexible (5). For [d.sub.v] = 0 the specification collapses into conventional GARCH model, whereas for [d.sub.v] = 1 model becomes an Integrated GARCH. The process is reduces to ARFIMA model when [h.sub.t] a constant.

In addition to autoregressive time series models, we also consider a univariate unobserved components model (UCM) of inflation, also known as local level model, having the following state space form (Harvey 1989; Commandeur, Koopman 2007),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

Where [[mu].sub.t] is time varying unobserved level, [[epsilon].sub.t] and [[eta].sub.t] are i.i.d. disturbances of measurement and transition equations. This model allows us to decompose inflation in two parts as level mt long run component and irregular components [[epsilon].sub.t] and [[eta].sub.t]. The second equation in (6) known as transition equation indicates that unobserved level of inflation is time varying and assumed to follow first order random walk process over time. The ratio between variances of the level and irregular components [[sigma].sup.2.sub.[eta]]/[[sigma].sup.2.sub.[epsilon]] is called signal to noise ratio. After maximum likelihood estimation of the parameter level [[mu].sub.t] is obtained through a Kalman filter. This process is also known as smoothing or signal extraction.

4. Empirical results

4.1. Model estimations

In this section, the models defined in the previous section are estimated. First we start from the neural network model. The estimation of the ANNs includes two steps. Training stage involves an iterative procedure where optimum hidden node and output node weights are adjusted by means of neural network algorithm. After the training, the model can be used to produce fitted or forecasted values. The sample period up to 2007:12 is accepted as the training set for ANN model, the remaining 24 observations are reserved to testing set.

We use backpropagation techniques with an adaptive learning rate algorithm to train the model to specified level of convergence. As a standard procedure, [[pi].sub.t] is mapped linearly into the range 0 and 1 to adjust for the logistic activation function (6). Our model is trained until the mean square error, the mean squared error between the actual and the fitted values of the output of the ANN, lies below the convergence criterion. As previously stated, determining the number of input and output nodes of the network is very important to produce good forecasts. We conduct estimations to find out whether a change in the number of input and output nodes makes any effect on the training performance. For this purpose RMSE values obtained from the training set of the models that contain different combinations of input and hidden nodes are reported in Table 1. RMSE values decrease with the increase in the number of input and hidden nodes used in the estimation. RMSE is minimized when the model is estimated with 10 and 11 input nodes. Hence we retain the ANN model with 11 and 4 hidden nodes as a best performer in training to use for the out of sample forecast comparison.

After the ANN, we estimate RW and ARIMA models. We do not employ Box and Jenkins (1976) approach in the modeling. From the forecasting point of view it may be disadvantageous to select p and q arbitrarily large. A higher order model will produce lower error give the best fit in sample, but when the model is used for out of sample forecasting purpose, it is likely to produce worse forecast than the lower order model, since the mean square error of the forecasts errors will not affected by only the stationary variance of the model but also by errors arising from the estimation of the parameters of the model (Brockwell, Davis 2002). Therefore we estimate different combinations of ARIMA models with maximum lag order for p = 15 and q = 2 to overcome this problem. ARIMA (8, 0, 0) is found to be the best model selected by Akaike Information Criterion (See Table 2)7. The sum of autoregressive parameters is obtained as 0.834 confirming the persistent behavior of inflation over the investigation period. Diagnostic tests reported in Table 2 show that the model suffers from non-normality. They also present some evidence for the first order ARCH effect, indicating that the variance of the error term is not constant over time. However we do not find any sign of autocorrelation in the residuals.

The maximum likelihood estimation of the ARFIMA is shown in Table 3. As compared to RW and ARIMA models persistence of inflation remains low, however the model provides similar fit to the data in terms of log-likelihood and Akaike information criteria. The estimated long memory conditional mean parameter d is 0.334 and significantly different from zero. Since the parameter lies in the range 0 < d < 0.5, one can say that inflation in Turkey follows a long memory stationary process.

We also estimate FIGARCH model explained above with maximum likelihood method based on Skewed Student distribution, with 2.9092 degrees of freedom. On general, the parameter estimates of the FIGARCH are found to be significant and corroborate the existence of long memory effects as in the ARFIMA model. The inclusion of conditional heteroscedasticity in the residuals improves fit of the model as evidenced by AIC and log-likelihood values (see Table 4). The model also yields a very high and significant fractional difference parameter [d.sub.v] with 0.775, suggesting that conditional variance of the inflation is characterized by a near integrated GARCH process. The estimated parameter of the skewed student distribution [psi] known as the asymmetry parameter within the range of -1 < [psi] < 1 is found to be positive and significant supporting right skewed characteristics of the data. Another distribution parameter of the distribution v also confirms heavy-tailed characteristics of the inflation (8).

Finally we estimate unobserved components model of inflation based on (6). The results of the estimated state vector by maximum likelihood are presented in Table 5. Local level model of inflation passed all diagnostic tests except for normality of the residuals. However cumulative sum of squared residuals (CUSUM) suggest that the model is stable (see Figure A3). The parameter of the unobserved level of inflation is found to be significant, suggesting that underlying level of inflation follows time varying pattern that can be modeled through the first order random walk process (see Figure 2). Estimated state vector shows that inflation has an increasing trend with about 0.94% on monthly basis.

[FIGURE 2 OMITTED]

4.2. Evaluating forecasting accuracy

The forecasting accuracy of alternative models is compared in terms of both static and dynamic forecasts. A Static (one-step-ahead) forecast uses the actual values for lagged dependent variables to compute forecasted values of dependent variable. Therefore this technique is more convenient for short term forecasting purpose. On the other hand, a dynamic forecast (multi-step ahead) has ability to produce long term projected values in which forecasted values are generated through an iterative procedure, i.e. forecasted values computed in the earlier periods are used as independent variables (Diebold 2007; Makridakis et al. 1998). Out of sample forecasted values of inflation are computed for each of the models in static and dynamic context. Besides, to account for the impact of change in forecasting path on the accuracy of alternative models we consider different starting points for each forecasting horizon (See Table 6). Hence we allowed each forecasting to end. All estimations are conducted using the sample period up to 2007:12 (training set for ANN model), the remaining 24 observations are reserved to compare out of sample forecasting accuracy (testing set for ANN model).

In order to evaluate the forecasting accuracy of the models, we consider several evaluation criterions. First we compute Mean Absolute Error (MAE) and Root-Mean Squared Error (RMSE) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

where [[??].sub.t] and [[pi].sub.t] represents forecasted and actual values of inflation respectively; h denotes forecast horizon h = 6, 12, 18, 24. In addition, scale invariant Theil inequality statistics (Theil) is also computed to evaluate forecasting performance (Pindyck, Rubinfeld 1998).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

Theil statictics is always between zero and one, where zero corresponds to perfect fit. Theil inequality statistics can be decomposed into three parts by following two steps. First, mean squared forecast error is divided into three parts:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] denote the means and standard deviations of [[??].sub.t] and [??], and r represents the correlation between actual and forecasted values. Dividing each part of (8) by the mean squared error itself results in the following decomposition of the Theil inequality statistics,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

The first term is the bias proportion, which shows contribution of the difference in means between the forecasted and actual series to the Theil inequality. The second term is known as the variance proportion, it measures the difference in variance between actual and forecasted values. The last term, the covariance proportion shows remaining unsystematic errors in the forecast. Forecasts are considered as good when the bias proportions are found relatively small compared to covariance proportions. Theil inequality statistics should be mostly explained by the covariance proportions (Pindyck, Rubinfeld 1998).

The criterions described above are computed with different horizons to evaluate out-of-sample forecasting accuracy of the alternative models. The findings show that linear models have the lowest RMSE values at shorter forecast horizon, however as forecast horizon gets longer ANN and UCM performs better. Forecasting accuracy statistics are illustrated in Table 6. Random walk without drift model is the worst performer in terms of both static and dynamic forecasts, since it presents highest value of RMSE, MAE and Theil-Inequality statistics at each forecasting horizons. When the one-step ahead forecasts are considered, ARFIMA model turns out be the best performer according to MAE and RMSE criterions (see Figure 3). ARIMA and FIGARCH models are also provide better fit than UCM and ANN at six months. However the performance of UCM has improved considerably with the increase in forecasting horizons, it becomes the second best model after the ARFIMA for the horizons longer than six months.

[FIGURE 3 OMITTED]

The superiority of UCM is more pronounced in terms of dynamic forecasts. The statistics of ANN is very close to that of UCM model up to 18 months. Hence ANN turns out to be the second best model in terms of tracking inflation. In autoregressive models the bias proportion accounts for more than 50 percentage of Theil inequality indicating the presence of systematic error between actual and forecasted values. This also corroborates the evidence on the lower quality of forecasts produced by the other statistics. Overall, the UCM provides superior long-term forecasts compared to the other techniques.

5. Conclusions

The stability of prices, as an important indicator of overall economic performance is one of the main objectives of monetary policy. In recent years, inflation targeting, has been implemented by most of the central banks around the world, as a policy tool of maintaining price stability. However, maintaining price stability over a period requires forward-looking approaches due to the lags and dynamic structure of price movements. Therefore, predicting and forecasting inflation has emerged as a crucial factor to be placed in monetary policy decisions for almost every monetary decision making process. In this paper we evaluate inflation forecasting accuracy of alternative time series models. For this purpose, static and dynamic forecasts are produced and the accuracies are compared by RMSE, MAE and Theil inequality measures. Static forecast uses the actual values in forecasting, and likely to produce more accurate results for short term forecasting. However, a dynamic forecast employs iterated forecast values and more convenient for long term projections.

We find that at earlier forecast horizons conventional autoregressive models, especially ARIMA and ARFIMA, provide better forecasting performance. However when the horizon gets longer, UCM turns out to be the best performer. The accuracy of ANN model has improved considerably with an increase in the forecast horizon. The superiority of UCM over the alternative models suggests that inflation in Turkey has a time varying pattern and linear models fail to track underlying level of inflation.

These findings indicate that the superiority of the models depends on the forecasting horizon. Inflation targeting policies require long term predictability of price changes, policy makers can consider UCM and ANN models as serious candidates that are able to better track the future values of inflation than the conventional autoregressive models.

doi:10.3846/16111699.2011.620135

APPENDIX

Table A1. The descriptive statistics for [[pi].sub.t] Whole Period After 2003 Mean 3.048 0.703 Median 2.877 0.713 Maximum 27.004 4.316 Minimum -2.647 -2.647 Std. Dev. 2.542 1.171 Skewness 2.893 0.273 Kurtosis 26.186 4.236 Jarque-Bera 7971.366 6.391 Probability 0.000 0.041 Sum 1021.165 59.028 Sum Sq. Dev. 2157.651 113.908 Observations 335 84 ADF -9.680 -6.159 (0.000) (0.000) Note: The critical values for ADF test are -3.98, - 3.42, and - 3.13 at 1%, 5%, and 10% significance level respectively

[FIGURE A1 OMITTED]

[FIGURE A2 OMITTED]

[FIGURE A3 OMITTED]

Received 26 January 2011; accepted 28 April 2011

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(1) Some studies find evidence in favour of neural network models (see for example Binner et al. 2004; Nakamura 2005; Panda, Narasimhan 2007) whereas the others not (Faraway, Chatfield 1998; Stock, Watson 1998; Heravi et al. 2004).

(2) Series available on IFS database is not originally seasonally adjusted. In order to obtain adjusted CPI series we employ Tramo-Seats method.

(3) See Zhang et al. (1998) for detailed literature survey on forecasting with ANN.

(4) See Figure A2 in Appendix for the general architecture of the feed-forward neural network model.

(5) For detailed information on the estimation of FIGARCH model see Chung (1999).

(6) In order to obtain the data on normalized inflation [[pi].sup.norm.sub.t], we use the following conventional formula (Mehrotra et al. 1996; Panda, Narasimhan 2007) [[pi].sup.norm.sub.t] = ([[pi].sub.t] - [[pi].sup.min.sub.t]) /([[pi].sup.max].sub.t] - [[pi].sup.min.sub.t]) * [[pi].sup.min.sub.t] and [[pi].sup.max.sub.t] denote minimum and maximum values of inflation respectively.

(7) An automated ARIMA model selection procedure written in RATS is used to obtain AIC for all combinations of ARIMA models.

(8) For detailed information on the functional form of skewed student distribution see Lambert and Laurent (2000).

A. Nazif CATIK is an Assistant Professor at Ege University, Turkey, with Ph.D. His scientific interests are related to the nonlinear analysis of macro data and time series. He has written in cited and indexed journals like Emerging Markets Finance and Trade, Applied Economics Letters, The Journal of Developing Areas, Economics Bulletin, while he has been reviewer in various journals such as METU Development Studies, Journal of Economic Studies. He has published papers in national and international conferences on time series analysis and robust statistics.

Mehmet KARACUKA is an Assistant Professor at Ege University, Turkey, with Ph.D. in Economics, and he is teaching Microeconomics and Applied Econometrics. His research interests are Applied Econometric Methods and Industrial Organization. He has written in cited journals like Telecommunications Policy, Applied Economics Letters, The Journal of Developing Areas; while he has been reviewer in different national and international journals such as The Journal of Developing Areas, The Service Industries Journal.

A. Nazif Catik [1], Mehmet Karacuka [2]

Department of Economics, Ege University, 35040 Bornova / Izmir, Turkey E-mails: [1] a.nazif.catik@ege.edu.tr (corresponding author); [2] mehmet.karacuka@ege.edu.tr

Table 1. Training performance of the alternative ANN specifications Number Number RMSE Number Number of RMSE of Input of Hidden of Input Hidden Nodes Nodes Nodes Nodes 1 1 0.0833 7 1 0.0430 1 2 0.0641 7 2 0.0425 1 3 0.0649 7 3 0.0424 1 4 0.0649 7 4 0.0398 2 1 0.0441 8 1 0.0429 2 2 0.0436 8 2 0.0424 2 3 0.0447 8 3 0.0411 2 4 0.0447 8 4 0.0396 0.0443 0.0415 3 1 0.0559 9 1 0.0427 3 2 0.0544 9 2 0.0424 3 3 0.0541 9 3 0.0412 3 4 0.0447 9 4 0.0412 4 1 0.0544 10 1 0.0426 4 2 0.0529 10 2 0.0412 4 3 0.0447 10 3 0.0400 4 4 0.0447 10 4 0.0387 5 1 0.0447 11 1 0.0425 5 2 0.0447 11 2 0.0424 5 3 0.0428 11 3 0.0400 5 4 0.0424 11 4 0.0387 6 1 0.0429 12 1 0.0436 6 2 0.0424 12 2 0.0436 6 3 0.0424 12 3 0.0413 6 4 0.0388 12 4 0.0412 Note: RMSE calculations are based on the normalized values of the output variable Table 2. Maximum likelihood estimation of ARIMA (8, 0 ,0) model: 1982(2)-2009(12) Coefficient S. E. [c.sub.0] -0.036 (0.639) [[PHI].sub.1] 0.450 (0.054) [[PHI].sub.2] 0.092 (0.060) [[PHI].sub.3] -0.071 (0.060) [[PHI].sub.4] 0.018 (0.059) [[PHI].sub.5] 0.177 (0.059) [[PHI].sub.6] 0.011 (0.060) [[PHI].sub.7] 0.047 (0.060) [[PHI].sub.8] 0.110 (0.054) Log-Likelihood -692.510 AIC 4.194 [[sigma].sup.2] 3.667 Descriptive Statistics for Residuals Normality test [chi square] (2) 30.090 [0.0000] ARCH test F(1.324) 3.559 [0.0601] Autocorrelation test 32.146 [0.2686] [c.sup.2] (28) Note: *, ** and *** indicate significant at 10, 5 and 1% respectively. Numbers in brackets represents the probabilities of residual tests Table 3. Maximum likelihood estimation of ARFIMA (1, d, 0) model: 1982(2)-2009(12) Coefficient S.E. d 0.334 (0.059) [[PHI].sub.1] 0.635 (0.061) Log-Likelihood -698.424 AIC 4.188 [[sigma].sup.2] 3.805 Descriptive Statistics for Residuals Normality test [chi square](2) 57.39 [0.0000] ARCH test F(1.331) 0.42479 [0.5150] Autocorrelation test 43.628 [0.1247] [chi square](28) Note: *, ** and *** indicate significant at 10, 5 and 1% respectively. Numbers in brackets represents the probabilities of residual tests Table 4. Maximum likelihood estimate of ARMA-FIGARCH model: 1982(2)-2009(12) Coefficient S. E. Constant (Mean Eq.) 3.334 *** (0.478) [[PHI].sub.1] 0.784 *** (0.051) Constant (Variance Eq.) 4.880 *** (0.793) [d.sub.v] 0.775 *** (0.207) [[alpha].sub.1] 0.676 *** (0.022) [[beta].sub.1] 0.290 *** (0.040) [psi] 0.226 * (0.119) [delta] 2.909 *** (0.352) Log-Likelihood -572.813 AIC 3.46754 Skewness (Y) 2.89347 Kurtosis (Y) 26.18608 Note: *, ** and *** indicate significant at 10, 5 and 1% respectively. The robust standard given in parentheses. The model is estimated with maximum likelihood method based on Student distribution, with 2.9092 degrees of freedom Table 5. Univariate unobserved components model of inflation: 1982(2)-2009(12) Hyperparameters Parameter (q-ratio) Level [[sigma].sup.2.sub.[eta]] 0.269 (0.086) Irregular 3.112 (1.000) [[sigma].sup.2.sub.[epsilon]] Estimated coefficients Coefficient t-values of the final state vector Level [[mu].sub.t] 0.947 (0.019) Diagnostic Statistics Coefficient p-values [R.sup.2] 0.342 Q 11.362 (0.07) Normality 88.460 (0.00) H 0.894 Notes: [R.sup.2] is the coefficient of determination, Q is Box-Ljung serial correlation statistic, based on the first 12 residual autocorrelations and tested against a chi-square distribution with six degrees of free-dom. H is a basic non-parametric test of heteroscedasticity, is a two-sided F test centered around unity. A low value indicates a decrease in the variance over time. The critical values for this test at the 5% level are 0.56 and 1.77 Table 6. Forecasting accuracy of alternative time series models Out-of-Sample Static From 2009:06 to 2009:12 RW ARIMA ARFIMA FIGARCH UCM Mean Absolute Error 0.822 0.394 0.392 0.454 0.573 Root Mean Square Error 0.949 0.574 0.430 0.595 0.669 Theil Inequality Measure 0.342 0.251 0.220 0.259 0.392 Bias 0.751 0.338 0.001 0.254 0.456 Variance 0.035 0.147 0.233 0.011 0.000 Covariance 0.213 0.515 0.766 0.735 0.544 From 2009:01 to 2009:12 RW ARIMA ARFIMA FIGARCH UCM Mean Absolute Error 1.161 0.767 0.681 0.830 0.732 Root Mean Square Error 1.303 1.004 0.817 1.076 0.972 Theil Inequality Measure 0.525 0.488 0.471 0.514 0.602 Bias 0.536 0.228 0.054 0.097 0.191 Variance 0.013 0.025 0.138 0.005 0.006 Covariance 0.451 0.747 0.808 0.898 0.804 From 2008:06 to 2009:12 RW ARIMA ARFIMA FIGARCH UCM Mean Absolute Error 1.482 1.232 1.069 1.255 1.077 Root Mean Square Error 1.929 1.789 1.519 1.726 1.493 Theil Inequality Measure 0.674 0.694 0.683 0.685 0.705 Bias 0.448 0.307 0.169 0.158 0.001 Variance 0.068 0.092 0.169 0.021 0.052 Covariance 0.485 0.601 0.662 0.821 0.947 From 2008:01 to 2009:12 RW ARIMA ARFIMA FIGARCH UCM Mean Absolute Error 1.277 1.074 0.956 1.088 1.027 Root Mean Square Error 1.743 1.608 1.377 1.558 1.392 Theil Inequality Measure 0.53 0.541 0.520 0.522 0.539 Bias 0.408 0.24 0.101 0.138 0.001 Variance 0.106 0.151 0.236 0.036 0.05 Covariance 0.486 0.609 0.663 0.826 0.949 Out-of-Sample Static Out-of-Sample Dynamic From 2009:06 to 2009:12 ANN RW ARIMA ARFIMA Mean Absolute Error 0.491 2.343 1.795 1.375 Root Mean Square Error 0.640 2.396 1.852 1.455 Theil Inequality Measure 0.297 0.562 0.498 0.442 Bias 0.036 0.956 0.940 0.893 Variance 0.000 0.044 0.056 0.103 Covariance 0.964 0.000 0.004 0.004 From 2009:01 to 2009:12 ANN RW ARIMA ARFIMA Mean Absolute Error 0.830 2.750 2.099 1.728 Root Mean Square Error 1.100 2.856 2.216 1.878 Theil Inequality Measure 0.546 0.688 0.633 0.600 Bias 0.010 0.927 0.898 0.847 Variance 0.020 0.073 0.082 0.138 Covariance 0.970 0.000 0.021 0.015 From 2008:06 to 2009:12 ANN RW ARIMA ARFIMA Mean Absolute Error 1.277 3.032 2.211 1.942 Root Mean Square Error 1.793 3.263 2.483 2.251 Theil Inequality Measure 0.724 0.734 0.684 0.674 Bias 0.105 0.864 0.783 0.729 Variance 0.014 0.132 0.134 0.220 Covariance 0.881 0.005 0.082 0.051 From 2008:01 to 2009:12 ANN RW ARIMA ARFIMA Mean Absolute Error 1.109 2.544 1.858 1.642 Root Mean Square Error 1.616 2.925 2.230 2.023 Theil Inequality Measure 0.549 0.640 0.602 0.584 Bias 0.09 0.756 0.546 0.505 Variance 0.025 0.127 0.159 0.279 Covariance 0.885 0.117 0.295 0.217 Out-of-Sample Dynamic From 2009:06 to 2009:12 FIGARCH UCM ANN Mean Absolute Error 2.460 0.416 0.452 Root Mean Square Error 2.511 0.513 0.529 Theil Inequality Measure 0.573 0.281 0.254 Bias 0.960 0.033 0.095 Variance 0.040 0.967 0.903 Covariance 0.000 0.000 0.002 From 2009:01 to 2009:12 FIGARCH UCM ANN Mean Absolute Error 2.853 0.657 0.741 Root Mean Square Error 2.954 0.834 0.957 Theil Inequality Measure 0.695 0.487 0.487 Bias 0.933 0.142 0.351 Variance 0.065 0.858 0.643 Covariance 0.003 0.000 0.005 From 2008:06 to 2009:12 FIGARCH UCM ANN Mean Absolute Error 3.067 0.968 1.072 Root Mean Square Error 3.292 1.349 1.472 Theil Inequality Measure 0.734 0.667 0.650 Bias 0.868 0.202 0.335 Variance 0.107 0.798 0.656 Covariance 0.025 0.000 0.009 From 2008:01 to 2009:12 FIGARCH UCM ANN Mean Absolute Error 2.508 1.016 1.475 Root Mean Square Error 2.938 1.345 1.767 Theil Inequality Measure 0.648 0.599 0.581 Bias 0.715 0.029 0.297 Variance 0.087 0.000 0.244 Covariance 0.198 0.971 0.459

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Author: | Catik, A. Nazif; Karacuka, Mehmet |
---|---|

Publication: | Journal of Business Economics and Management |

Article Type: | Report |

Geographic Code: | 7TURK |

Date: | Apr 1, 2012 |

Words: | 6871 |

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