Modeling Inflation in Bangladesh

Mohammad Lutfor Rahman^*, Mynul Islam, Manik Roy

Institute of Statistical Research and Training (ISRT), University of Dhaka, Dhaka, Bangladesh.

DOI: 10.4236/ojs.2020.106056                

Abstract

Inflation has a substantial impact on an economy because it affects the financial value of money and stability in the economy. Government and non-govern- ment policies might be hindered due to the excessive rate of inflation. This paper aims to model and forecast inflation by the Box-Jenkins autoregressive integrated moving average (ARIMA) technique using annual time series data on inflation from 1987 to 2017 in Bangladesh. It is found that ARIMA (2, 1, 0) model is the best optimal to forecast inflation for a period of up to eight years. Graphical tools, as well as theoretical tests such as Ljung-Box, Shapiro-Wilk, and runs tests have been used in the model diagnostics.

Keywords

ARIMA, Inflation, Forecasting, Model Validity, Model Diagnostics, Bangladesh

1. Introduction

Inflation which refers to the purchasing power of money is one of the most perpetual economic challenges in the world, particularly for the developing economies [1] [2]. Inflation has a substantial impact on the economy of a country because high inflation distorts level of price, discourages investment and hinders economic development. Controlling inflation or maintaining low inflation is critical to protecting the purchasing power of the poor, in particular food-price, in developing countries [3]. Inflation badly affects many economic indicators such as money supply, tax revenues, government expenditures, exports imports, gross domestic products (GDP), exchange rate, stock and bond returns, and others [4]. Inflation causes the devaluation of savings [5]. It is perceived that lower the inflation better the financial management. Because long term plans become hard to achieve when there is greater uncertainty in future inflation [6]. Inflation may be a non-ignorable problem if it goes out of control in a developing country similar to Bangladesh. Therefore, it is essential to keep an eye on inflation rate in any country for better financial management, to preserve wealth and greater stability in the economy. Proper investigation and steps may be helpful to control inflation at a tolerable level. Existing research on this economic variable is inadequate in Bangladesh. This paper is an endeavor to forecast inflation using univariate long term data of inflation from 1987 to 2017 in Bangladesh. The R programming language (version 4.0.0) has been used for data analysis purposes.

The rest of the study has been organized as follows: Section 2 demonstrates the literature review, Section 3 describes data and methodology, Section 4 presents the results and discussions, and the last Section 5 has drawn conclusions of the study.

2. Literature Review

In history, many studies were carried out on the comparative precision of different models of inflation forecasting. Yusif et al. [7] used artificial neural network modeling approach for forecasting inflation. Hafer and Hein [8] compared interest rate based forecasting model and univariate time series model based on monthly data from the United States, Belgium, Canada, England, France and Germany and found time series forecast of inflation model producing equal or lower forecast errors and has unbiased predictions than the interest rate based forecasts. Sun [9] combined short-term model with an equilibrium correction model for projecting core inflation using monthly data during 1995 to 2003 in Thailand. For Indonesia, Ramakrishnan and Vamvakidis [10] estimated a multivariate model to identify the leading indicators that have predictive information on future inflation using quarterly data from 1980 to 2000.

Vector autoregressive (VAR) models have been employed for forecasting inflation by Lack et al. [11] in Switzerland; Callen and Chang [12] in India; Kelikume and Salami [13], Inam [14] in Nigeria; and Younus and Roy [15] in Bangladesh. The generalized autoregressive conditional heteroscedasticity (GARCH) models were investigated for inflation forecasting by Nyoni and Nathaniel [6] in Zambia; Fwaga et al. [16] in Kenya; Ngailo et al. [17] in Tanzania; and Banerjee [18] for 41 developed and developing countries for the time period 1958-2016. Akhtaruzzaman [19] used cointegration and vector error correction modeling (VECM) technique in Bangladesh; and Bokil and Schimmelpfennig [20] employed three empirical approaches based on monthly data to forecast inflation in Pakistan.

A vast majority studies across the world used Box-Jenkins ARIMA technique for modeling inflation. For instance, Salam et al. [21] in Pakistan, Habibah et al. [4] for SAARC countries; Faisal [22] in Bangladesh, Meyler et al. [23] in Ireland; Iftikhar [24] in Pakistan; Okafor and Shaibu [25], John and Patrick [26], Mustapha and Kubalu [27], Popoola et al. [28] in Nigeria; Jere and Siyanga [2] in Zambia; Islam [29] and Habibah et al. [4] in Bangladesh found ARIMA model as the better model for forecasting inflation. By augmenting seasonal component some studies found seasonal autoregressive integrated moving average (SARIMA) model as the best optimal. For example, Akhter [30] in Bangladesh, Out et al. [31] in Nigeria, and Lidiema [32] in Kenya used SARIMA for modeling and forecasting inflation.

Also, some authors implemented several methods simultaneously for comparison purposes. Nyoni and Nathaniel [6] used ARMA, ARIMA, and GARCH mdoels for forecasting inflation in Nigeria based on time series data on inflation rates from 1960 to 2016; of which they found ARMA (1, 0, 2) as the best optimal. Pincheira and Gatty [33] used FASARIMA, ARIMA, SARIMA and FASARIMAX methods for forecasting inflation of 18 Latin American and 30 OECD countries. Lidiema [32] found that SARIMA model was better model than the Holt-winter’s triple exponential smoothing for forecasting inflation in Kenya. Ingabire and Mung’atu [34] found ARIMA (3, 1, 4) model better than VAR model for forecasting Rwanda’s inflation rate.

Akhter [30] employed seasonal auto-regressive integrated moving average (SARIMA) model to forecast short-term inflation rate of Bangladesh using the monthly consumer price index (CPI) from January 2000 to December 2012. Though Islam [29] and Habibah et al. [4] attempted recently to forecast inflation by ARIMA (1, 0, 0) and ARIMA (3, 0, 0) models respectively, but their prediction slides the reality as we found the actual inflation rate obtained from Bangladesh Bureau of Statistics (BBS) differ substantially in the subsequent years. In this study, we thrive for a precise ARIMA model for forecasting inflation in Bangladesh.

3. Data and Methodology

To model inflation rate in Bangladesh, long term univariate time series data on inflation obtained from the World Bank (2019) from 1987 to 2017 were used. There are several approaches for modelling time series data with seasonal patterns. The autoregressive integrated moving average (ARIMA) model developed by Box and Jenkins [35] and Box and Tiao [36] is one of the frequently appeared approaches for handling time series data.

The ARIMA model that is usually denoted as ARIMA (p, d, q) addresses time dependence in several ways. Firstly, the time series are d-differenced to make the series stationary. When d = 0, the series is considered as stationary and modelled directly, and if d = 1, the differences between consecutive observations are modelled. Secondly, the time dependence of the stationary process X_t is modelled by incorporating p autoregressive models. The equation for order p is:

$X_t=C+{{\displaystyle \sum }}^{\text{}}{\varphi }_i{X}_{t-i}+Z_t$ (1)

where C is a constant, ${\varphi }_i$ is the parameter of the model, X_t is the observed value at time t, Z_t represents random error. Thirdly, q stands for moving-average term. It includes the observations of the previous random errors. The equation for moving average of order q is:

$X_t={{\displaystyle \sum }}^{\text{}}{\theta }_i{Z}_{t-i}+Z_t$ (2)

θ_i is the model parameter, Z_t is the white noise or error term. Finally, we obtain the ARIMA model by combining Equations (1) and (2). Thus, the usual form of the ARIMA models can be presented as follows:

$X_t=C+{{\displaystyle \sum }}^{\text{}}{\varphi }_i{X}_{t-i}+{{\displaystyle \sum }}^{\text{}}{\theta }_i{Z}_{t-i}+Z_t$ (3)

In the current study, the stationarity of the data was tested by augmented Dickey-Fuller (ADF) test. The candid ARIMA model was selected by judging the values of the criteria Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). The Shapiro-Wilk test and runs test were used for checking the normality and randomness of the residuals.

Generally, ARIMA models use the back-shift operator B which is defined as $B_K\left(X_t\right)=X_{t-k};t>k,t,k\in N$, where k is the index representing how many times back-shift operator B is applied to time series X_t characterized by time interval t, and N is the total number of time intervals. Using the following notations

$\Phi \left(z\right)=1+{{\displaystyle \sum }}^{\text{}}{\varphi }_i{z}^i;{\varphi }_p\ne 0$

$\Theta \left(z\right)=1+{{\displaystyle \sum }}^{\text{}}{\theta }_i{z}^i;{\theta }_q\ne 0$

Equation (3) can be written as


$\Phi \left(B\right){\left(1-B\right)}^d{X}_t=C+\Theta \left(B\right)Z_t$

To determine an appropriate model for a given time series data, it is essential to figure out the autocorrelation function (ACF) and partial autocorrelation function (PACF) analysis, which exhibit how the observations in a time series are interrelated. The plot of ACF helps to determine the order of moving average terms, and the plot of PACF helps to determine autoregressive terms.

In the current study, the stationarity of the data was tested by augmented Dickey-Fuller (ADF) test. The candid ARIMA model was selected by judging the values of the criteria: Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). The Shapiro-Wilk test and runs test were used for checking the normality and randomness of the residuals.

4. Results and Discussions

Data Inspection

The first step of time series analysis is to inspect the graph of the data [37]. The aim of inspecting the plot of the data is to observe if there is any visible pattern in the data i.e. to observe whether there is any seasonality in the data.

The first plot in Figure 1 shows the inflation rate of Bangladesh since 1987 until 2017. It is observed from the graph that there is no apparent pattern and seasonality in the data. Also, no sign of stationary of the data is observed by this plot. The second plot which is the second difference of inflation rate of Bangladesh also shows no apparent seasonality in the data. Thus the graphs in Figure 1 reveal that there is no seasonality and trend in the data.

Figure 1. Plot of inflation rate of Bangladesh.

Checking Stationarity

The second step of time series model building is to check stationarity of the data which can be done by the augmented Dicky Fuller (ADF) test [37]. The non-stationarity of the time series data corresponds null hypothesis H₀ in the ADF test. Table 1 indicates that original data as well as first order difference series are not stationary (p-value > 0.05). However, after taking second order difference, the series becomes stationary (p-value < 0.05).

Autocorrelation and Partial Autocorrelation Functions

The third step in time series analysis is to find the order of autoregressive (AR) and moving average (MA) models by ACF and PACF [37]. The ACF and PACF are two important functions for checking autocorrelations of different lags in the data. Significant autocorrelation of any time lag of the series indicates the order of the moving average model and significant partial autocorrelation indicates the order of the autoregressive model. However, orders of the moving average and autoregressive models are approximate, further analysis is required to confirm the orders.

In Figure 2 we notice that only first order lag is significant in ACF plot and first three order lag is significant in PACF plot. Thus, our tentative model is ARIMA (3, 2, 1).

Model Determination

Selection of the best model is a crucial part of predicting inflation. There are some criteria such as AIC, BIC, and log likelihood for selecting the best model. The lower AIC, BIC and higher log likelihood values might indicate the probable best model. In Table 1, the augmented Dicky Fuller test found that data became stationary after taking second difference and later ACF and PACF helps finding the temporary orders of MA and AR models and thereby the tentative model was ARIMA (3, 2, 1). To find the competitive models, we can hover around the tentative ARIMA (3, 2, 1) model and compute criteria AIC and BIC values. We compare AIC and BIC values for different ARIMA models in Table 2 to choose the best model.

In Table 2 we observe that the lowest values of AIC and BIC are 144.0352 and 149.5044 respectively corresponding to ARIMA (2, 2, 1) and thus ARIMA (2, 2, 1) would be selected tentatively for inflation prediction.

Figure 2. Graph of ACF and PACF.

Table 1. Augmented Dicky-Fuller test.

Table 2. Model selection based on AIC and BIC.

Table 3 presents the coefficients, standard errors, and 95% confidence intervals of the ARIMA (2, 2, 1) model. Thus, the fitted model ARIMA (2, 2, 1) can be written as follows:

$\left(1-{\varphi }_{1}B-{\varphi }_2{B}^2\right){\left(1-B\right)}^2{X}_t=\left(1+{\theta }_{1}B\right)Z_t$

$\left(1+0.3614B+0.4750B^2\right){\left(1-B\right)}^2{X}_t=\left(1-B\right)Z_t$

$\left(\text{1}+0.\text{3614}B+0.4750B^2\right)\left(1-B\right)X_t=Z_t$

where B denotes back shift operator, $X_t$ is the time series data, $Z_t$ represents white noise. Due to having an estimate as −1.000 corresponding to the coefficient of MA1, the ARIMA (2, 2, 1) reduces to ARIMA (2, 1, 0). Therefore, ARIMA (2, 1, 0) which has lower AIC (141.1012) and BIC (145.3048) values than that of ARIMA (2, 2, 1) has been taken into consideration as the final model for forecasting inflation in Bangladesh. Table 4 presents the coefficients, standard errors, and 95% confidence intervals of the ARIMA (2, 1, 0) model.

Finally, the selected ARIMA (2, 1, 0) model can be written algebraically as follows:

$\left(1-{\varphi }_{1}B-{\varphi }_2{B}^2\right)\left(1-B\right)X_t=Z_t$

$\left(1+0.3776B+0.4915B^2\right)\left(1-B\right)X_t=Z_t$

where the interpretations of notations remain as before.

Model Diagnostics

Randomness and independence of residuals are two important assumptions in modeling. The ACF plot in Figure 3 shows that residuals are scattered both sides of zero line without making any pattern. Therefore, it is believed that residuals are randomly distributed. Randomness of the residuals can also be checked by Wald-Wolfowitz runs test. The large p-value (>0.05) in Table 5 implies that the null hypothesis of randomness is not rejected and thus residuals are random in nature. Further, we notice in Figure 3 under the PACF plot that none of the points are outside the significance line which proves the independence of residuals. The Ljung Box test in Table 6 supports the evidence of independence further by providing larger p-value (>0.05) and thereby not rejecting the null hypothesis of independence.

Figure 3. Diagnostic plot of residuals.

Table 3. Coefficients of ARIMA model.

Table 4. Coefficients of ARIMA (2, 1, 0) model.

Table 5. Wald-Wolfowitz runs test for randomness.

Table 6. Ljung Box test.

The normality assumption of residuals can be checked by a Q-Q plot. The Q-Q plot shows that the points are roughly lie on a straight line which ensures the normality assumption of residuals (Figure 4). Further, Shapiro-Wilk normality test (Table 7) has p-value greater than 0.05 which leads not to reject the null hypothesis at 5% level of significance. Thus, it is concluded from the results that residuals are normal.

Plot of Fitted Versus Actual Values

The actual and fitted values of inflation rate have been presented in Figure 5 which indicates that selected model performs well in terms of prediction. Although there exists some discrepancy between fitted and actual values, it might be reasonable to carry on for prediction.

Forecasted and Real Values of Inflation

Using ARIMA (2, 1, 0) model we have forecasted eight steps ahead values (from year 2018 until year 2025) of inflation in Bangladesh (Table 8).

In Table 9, we presented actual inflation data from Bangladesh Bureau of Statistics (BBS) where point to point inflation (%) means inflation calculated based on previous month and 12 months average inflation (%) means inflation calculated based on last 12 months. However, we notice overwhelmingly that forecasted and actual inflations closely match for the available years, particularly for year 2018, year 2019 and year 2020 (Table 8 and Table 9). As predicted and actual values are nearly close, the ARIMA (2, 1, 0) model and its predictability might be acceptable for predicting inflation for a developing economy, particularly for Bangladesh. However, it is noted that our model is differing from models found in studies by Islam [29] and Habibah et al. [4]. The main reasons could be use of different sets of data. Also, as noted by Stockton and Glassman [38], for purposes of forecasting, econometric models differ not only in their specification, but also in the quantity and quality of the information presumed to be available to the forecaster.

Table 7. Shapiro-Wilk normality test.

Table 8. Eight steps ahead prediction of inflation.

Table 9. Real inflation data in Bangladesh.

Source: Bangladesh Bureau of Statistics (BBS).

Figure 4. Q-Q plot of residuals.

Figure 5. Fitted versus real values of inflation rate.

5. Conclusion

In this paper, we attempted to build up a suitable model for predicting inflation rate and found ARIMA (2, 1, 0) is the best optimal for forecasting inflation in Bangladesh up to eight years (Table 8). Comparison between actual and predicted values of inflation (Table 9) shows the efficiency and diagnostics analyses (Figure 3, Figure 4, Table 5, Table 6) show the validity of our model. By using this model interested stakeholders can forecast inflation rate in Bangladesh and thereby the policymakers can make use of the forecasted inflation rate at the time of making various economic policies. This study can be a good reference for inflation forecasting in other developing countries similar to Bangladesh.

Acknowledgements

The authors would like to thank anonymous reviewer for valuable comments and suggestions that helped substantially to improve the final version of the paper.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

[1]	Kellison, S.G. (2006) The Theory of Interest. McGraw-Hill Education, New York.
[2]	Jere, S. and Siyanga, M. (2016) Forecasting Inflation Rate of Zambia Using Holt’s Exponential Smoothing. Open Journal of Statistics, 6, 363-372. https://doi.org/10.4236/ojs.2016.62031
[3]	Kinda, M.T. (2011) Modeling Inflation in Chad. Number 11-57. International Monetary Fund, Washington DC. https://doi.org/10.5089/9781455221011.001
[4]	Habibah, U., Bhutto, N.A. and Ghumro, N.H. (2017) Inflation Forecasting in SAARC Region Using ARIMA Models. Sukkur IBA Journal of Economics and Finance, 1, 38-58. https://doi.org/10.30537/sijef.v1i1.131
[5]	Arnold, E., Drager, L. and Fritsche, U. (2014) Evaluating the Link between Consumers’ Savings Portfolio Decisions, Their Inflation Expectations and Economic News. Technical Report, DEP (Socioeconomics) Discussion Papers, Macroeconomics and Finance Series.
[6]	Nyoni, T. and Nathaniel, S.P. (2018) Modeling Rates of Inflation in Nigeria: An Application of ARMA, ARIMA and GARCH Models.
[7]	Yusif, M.H., Eshun Nunoo, I.K. and Sarkodie, E.E. (2015) Inflation Forecasting in Ghana—Artificial Neural Network Model Approach. International Journal of Economics & Management Sciences, 4, 2162-6359. https://doi.org/10.4172/2162-6359.1000274
[8]	Hafer, R.W. and Hein, S.E. (1990) Forecasting Inflation Using Interest-Rate and Time-Series Models: Some International Evidence. Journal of Business, 63, 1-17. https://doi.org/10.1086/296480
[9]	Sun, T. (2004) Forecasting Thailand’s Core Inflation. Number 4-90. International Monetary Fund, Washington DC. https://doi.org/10.5089/9781451851427.001
[10]	Ramakrishnan, U. and Vamvakidis, A. (2002) Forecasting Inflation in Indonesia. Surface Science, 2-111. https://doi.org/10.5089/9781451853483.001
[11]	Lack, C., et al. (2006) Forecasting Swiss Inflation Using VAR Models. Technical Report, Swiss National Bank, Zürich.
[12]	Callen, M.T. and Chang, D. (1999) Modeling and Forecasting Inflation in India. Number 99-119. International Monetary Fund, Washington DC. https://doi.org/10.5089/9781451854152.001
[13]	Kelikume, I. and Salami, A. (2014) Time Series Modeling and Forecasting Inflation: Evidence from Nigeria. The International Journal of Business and Finance Research, 8, 41-51.
[14]	Inam, U. (2017) Forecasting Inflation in Nigeria: A Vector Autoregression Approach. International Journal of Economics, Commerce and Management, 5, 92-104.
[15]	Younus, S. and Roy, A. (2016) Forecasting Inflation and Output in Bangladesh: Evidence from a VAR Model. Research Department and Monetary Policy Department, Bangladesh Bank, Dhaka, Bangladesh.
[16]	Fwaga, S.O., Orwa, G. and Athiany, H. (2017) Modelling Rates of Inflation in Kenya: An Application of GARCH and EGARCH Models. Mathematical Theory and Modelling, 7, 75-83.
[17]	Ngailo, E., Luvanda, E. and Massawe, E.S. (2014) Time Series Modelling with Application to Tanzania Inflation Data. Journal of Data Analysis and Information Processing, 2, 49-59. https://doi.org/10.4236/jdaip.2014.22007
[18]	Banerjee, S. (2017) Empirical Regularities of Inflation Volatility: Evidence from Advanced and Developing Countries. South Asian Journal of Macroeconomics and Public Finance, 6, 133-156. https://doi.org/10.1177/2277978717695157
[19]	Akhtaruzzaman, M. (2005) Inflation in the Open Economy: An Application of the Error Correction Approach to the Recent Experience in Bangladesh. Policy Analysis Unit (PAU) of Bangladesh Bank.
[20]	Bokil, M. and Schimmelpfennig, A. (2005) Three Attempts at Inflation Forecasting in Pakistan, Vol. 5. International Monetary Fund, Washington DC.
[21]	Salam, M.A., Salam, S. and Feridun, M. (2007) Modeling and Forecasting Pakistan’s Inflation by Using Time Series ARIMA Models. Technical Report, Economic Analysis Working Papers.
[22]	Faisal, F. (2012) Forecasting Bangladesh’s Inflation Using Time Series ARIMA Models. World Review of Business Research, 2, 100-117.
[23]	Meyler, A., Kenny, G. and Quinn, T. (1998) Forecasting Irish Inflation Using ARIMA Models. Economic Analysis, Research and Publications Department, Central Bank of Ireland, Dublin, 1-48.
[24]	Iftikhar, N. (2013) Forecasting the Inflation in Pakistan: The Box-Jenkins Approach. World Applied Sciences Journal, 28, 1502-1505.
[25]	Okafor, C. and Shaibu, I. (2013) Application of ARIMA Models to Nigerian Inflation Dynamics. Research Journal of Finance and Accounting, 4, 138-150.
[26]	John, E.E. and Patrick, U.U. (2016) Short-Term Forecasting of Nigeria Inflation Rates Using Seasonal ARIMA Model. Science Journal of Applied Mathematics and Statistics, 4, 101-107. https://doi.org/10.11648/j.sjams.20160403.13
[27]	Mustapha, A.M. and Kubalu, A.I. (2016) Application of Box-Inflation Dynamics. Ilimi Journal of Arts and Social Sciences, 2, 127-142.
[28]	Popoola, O.P., Ayanrinde, A.W., Rau, A.A. and Odusina, M.T. (2017) Time Series Analysis to Model and Forecast Inflation Rate in Nigeria. Annals: Computer Science Series, 15, 174-178.
[29]	Islam, N. (2017) Forecasting Bangladesh’s Inflation through Econometric Models. American Journal of Economics and Business Administration, 9, 56-60. https://doi.org/10.3844/ajebasp.2017.56.60
[30]	Akhter, T. (2013) Short-Term Forecasting of Inflation in Bangladesh with Seasonal ARIMA Processes. MPRA Paper No. 43729. https://mpra.ub.uni-muenchen.de/43729
[31]	Otu, O.A., Osuji, G., Opara, J., Mbachu, H. and Iheagwara, A. (2014) Application of SARIMA Models in Modelling and Forecasting Nigeria’s Inflation Rates. American Journal of Applied Mathematics and Statistics, 2, 16-28. https://doi.org/10.12691/ajams-2-1-4
[32]	Lidiema, C. (2017) Modelling and Forecasting Inflation Rate in Kenya Using SARIMA and Holt-Winters Triple Exponential Smoothing. American Journal of Theoretical and Applied Statistics, 6, 161-169. https://doi.org/10.11648/j.ajtas.20170603.15
[33]	Pincheira, P. and Gatty, A. (2016) Forecasting Chilean Inflation with International Factors. Empirical Economics, 51, 981-1010.
[34]	Ingabire, J. and Mung’atu, J.K. (2016) Measuring the Performance of Autoregressive Integrated Moving Average and Vector Autoregressive Models in Forecasting Inflation Rate in Rwanda. International Journal of Mathematics and Physical Sciences Research, 4, 15-25.
[35]	Box, G.E., Jenkins, G.M. and Reinsel, G. (1970) Time Series Analysis: Forecasting and Control. Holden-Day, San Francisco, 498.
[36]	Box, G.E. and Tiao, G.C. (1975) Intervention Analysis with Applications to Economic and Environmental Problems. Journal of the American Statistical Association, 70, 70-79. https://doi.org/10.1080/01621459.1975.10480264
[37]	Shumway, R.H. and Stoffer, D.S. (2017) Time Series Analysis and Its Applications: With R Examples. Springer, Berlin. https://doi.org/10.1007/978-3-319-52452-8
[38]	Stockton, D.J. and Glassman, J.E. (1987) An Evaluation of the Forecast Performance of Alternative Models of Inflation. The Review of Economics and Statistics, 69, 108-117. https://doi.org/10.2307/1937907