Acta Universitatis Danubius. Œconomica, Vol 10, No 5 (2014)
Forecast Intervals for Inflation Rate
and Unemployment Rate in Romania
Mihaela Simionescu^{1}
Abstract: The main objective of this research is to construct forecast intervals for inflation and unemployment rate in Romania. Two types of techniques were employed: bootstrap technique (tpercentile method) and historical error technique (root mean square error method RMSE). The forecast intervals based on point forecasts of National Bank of Romania (NBR) include more actual values of quarterly inflation rate during Q1:2011Q4:2013. The proposed prediction intervals for quarterly inflation and unemployment rate contain the registered values. Considering as constant the error from previous year, we will build forecast intervals for annual inflation and unemployment rate based predictions provided by two anonymous experts on the horizon 20042015.All the forecast intervals for inflation rate based on first expert expectations included the actual values during 20042013.
Keywords: forecast intervals; point prediction; inflation rate; unemployment rate
JEL Classification: C51; C53
1 Introduction
The point forecasts did not provide any information regarding the degree of accuracy. On the other hand, the forecast intervals allow the evaluation of future uncertainty and the comparison between the forecasting methods, indicating the strategies to be applied for desired results.
The main aim of this paper is to construct forecast intervals for inflation and unemployment rate in Romania. Excepting some forecast intervals proposed by (Bratu, 2012, p. 146), in Romania prediction intervals for macroeconomic variables were not proposed. The most frequently used method for constructing forecast intervals is the historical errors method that supposes keeping constant an accuracy measure. The bootstrap method is also used when the distribution type of the sample is unknown.
A grid bootstrap was used to compute the median without bias by (Gospodinov, 2002, p. 86), but the evolution of the events should be characterized by a high persistence. The main disadvantage is the high volume of computations. (Guan, 2003, p. 79).
The sieve bootstrap technique allows for consistent estimators of conditional repartition in the case of nonparametric prediction intervals (Alonso, Pena and Romo, 2003, p. 182). In Romania there is a strong correlation between inflation and money, making us to believe if the money earning went too far (Croitoru, 2013, p. 6). Therefore, the inflation forecasting should be taken under control.
In this paper we used as forecasting methods to build prediction intervals the historical errors method and the bootstrap technique.
It would be necessary to continue the research and build some Bayesian forecast intervals and to compare the results with those obtained using usual methods. The paper continues with the methodological framework, providing different types of quarterly and annual forecast intervals for the two variables. Moreover, the intervals are constructed using as point forecasts the anticipations of two forecasters during 20042015. It seems that first expert generated better inflation forecast intervals than the second one.
2. Methodology
The prediction interval that uses the historical errors method considers that errors follow a normal distribution of zero mean and standard deviation that equals the root mean squared error (RMSE) of the histrorical forecasts. Given a certain level of significance (α), the forecast intervals are built as it follows:
_{ } (1)
_{ } the point prediction of the variable Y given at time t for the period (t+k)
_{ } quantile α/2 of normal distribution of zero mean and standard deviation equalled to1
The following multiple linear regression is considered:
_{ } (1)
Y vector (length: nx1)
X matrix (length: nxp)
_{ } vector of parameters (length: px1)
u vector of error terms (length: nx1)
_{ } residuals (_{ }=YX_{ })
_{ } parameter estimator (_{ })
The form of bootstrap model is:
_{ } (2)
Y* vector (length: nx1)
X* matrix (lenth: nxp)
_{ } parameter estimator (_{ })
_{ }random element
The selected sample is: _{ }. The random term from theoretical bootstrap process uses modified residuals:
_{ } (3)
The theoretical process is computed as:
_{ } (4)
i=1,2,..,n
border of iteration
_{ } resampled from _{ }
Given the random variable _{ } (_{ }), the interval for _{ } considers that _{ } has Student distribution (np degrees of freedom). For a level of confidence of (12_{ }) the interval is:
_{ } (5)
_{ } (6)
The percentilet bootstrap method is based on _{ }estimation. We build a bootstrap table, the values of _{ } are:
_{ } (7)
The percentilet forecast interval for _{ } is:
_{ } (8)
For the observation with number f of the exogenous variable X, the prediction is calculated using the model (Y dependent variable): _{ } . Having a normal distribution of the errors and the confidence interval (12_{ } the standard prediction interval is:
_{ } (9)
A prediction interval for _{ } is based on forecast error _{ }. The future value _{ }
_{ } (10)
It is based on a retrieval of an empirical distribution of the modified residuals. For replication b, the prediction error is:
_{ } (11)
_{ } (12)
The bootstrap forecast error is:
_{ } (13)
Given the empirical distribution of _{ } (_{ }, the percentiles are employed to determine the bootstrap prediction intervals (_{ }). The percentile prediction interval is:
_{ } (14)
For percentilet prediction interval, standard deviation estimator (_{ }) is :
_{ } (15)
_{ } (16)
The statistic _{ } is determined:
_{ } (17)
The percentilet prediction interval has the form:
_{ } (18)
3. Forecast Intervals
Using the quarter point forecasts and the prediction intervals provided by the National Bank of Romania, we built some forecast intervals based on historical errors methods by keeping constant the root mean square error (RMSE) of the previous 4 quarter. The horizon is 2011:Q12015:Q4.
Table 1. Forecast intervals for the inflation rate predicted by National Bank of Romania (2011:Q12015:Q4)
Quarter 
Forecast interval 
Forecast interval historical error method 
Point forecast 
Actual values 


Lower limit 
Upper limit 
Lower limit 
Upper limit 


T1:2011 
7.48 
7.95 
2.96 
17.96 
7.5 
1.013 
T2:2011 
7.93 
8.05 
0.73 
16.59 
7.93 
1.005 
T3:2011 
3.45 
3.58 
1.69 
8.59 
3.45 
0.990 
T4:2011 
3.14 
3.25 
0.99 
7.27 
3.14 
1.012 
T1:2012 
1.43 
2.52 
3.01 
7.81 
2.40 
1.010 
T2:2012 
1.35 
3.44 
5.02 
9.10 
2.04 
1.002 
T3:2012 
2.46 
5.20 
3.40 
14.06 
5.33 
1.022 
T4:2012 
1.57 
4.93 
4.42 
14.32 
4.95 
1.009 
T1:2013 
1.34 
5.33 
3.78 
14.16 
5.19 
1.003 
T2:2013 
1.04 
5.98 
2.91 
14.69 
5.89 
1.003 
T3:2013 
0.62 
4.77 
4.04 
11.06 
3.51 
0.988 
T4:2013 
0.81 
5.18 
2.18 
9.20 
3.51 
1.006 
T1:2014 
1.02 
7.93 
1.24 
6.60 
2.68 

T2:2014 
1.5 
3.5 
1.14 
6.70 
2.78 

T3:2014 
1.5 
3.5 
0.84 
7.00 
3.08 

T4:2014 
1.5 
3.5 
0.73 
7.11 
3.19 

T1:2015 
1.5 
3.5 
1.72 
6.12 
2.2 

T2:2015 
1.5 
3.5 
2.12 
5.72 
1.8 

T3:2015 
1.5 
3.5 
1.32 
6.52 
2.6 

T4:2015 
1.5 
3.5 
1.12 
6.72 
2.8 

Source: own computations
In the period from 2011 to 2013 only two forecast intervals of NBR include the actual values of inflation rate. The prediction intervals based on historical RMSE contain all the actual values during 20112013.
The variables with quarterly data that are used are: index of consumer prices that will be used in computing inflation rate, real exchange rate and unemployment rate on the period 2000:Q2014:Q4. The quarterly forecasts will be made for 20112014, after the aggregation of data for obtaining annual values. The Tramo/Seats method was applied to get seasonally adjusted data. The logarithm was applied for the index of consumer prices. The data in first difference was computed for unemployment rate and exchange rate (d_ur and d_er).
The seasonally adjusted and stationarized index of consumer prices is denoted by log_ip. The following valid model was obtained:
_{ } (19)
(std. error=0,08) (std. error=0,02)
(tcalc.=13.62) (tcalc.=11.24)
According to BreuschGodfrey test for the first lag, the errors are independent. The hypothesis of errors normal distrbution is checked using JarqueBera test and we do not have enought evidence to reject the normal repartition. According to White test, he errors are homoskedastic. The results of the application of these test are presented in Appendix 1.
For the seasonally adjusted and first differentiated quarterly unemployment rate (ur) an autoregressive model of order 1 is built, for which the errors are independent, homoskedastic and they follow a normal repartition (Appendix 2).
_{ } (20)
Table 2. Point forecasts and bootstraped forecast intervals using the linear regression model for inflation rate (%) (percentilet method) (horizon: 2011:Q12015:Q4)
Quarter 
Point forecasts 
Forecast intervals for inflation rate 
Actual values 



Intervals limits 


Q1:2011 
1.0114 
0.0245 
1.9983 
1.013 
Q2:2011 
1.0088 
0.0219 
1.9957 
1.005 
Q3:2011 
1.0059 
0.0190 
1.9928 
0.990 
Q4:2011 
1.0075 
0.0206 
1.9944 
1.012 
Q1:2012 
1.0044 
0.0175 
1.9913 
1.010 
Q2:2012 
1.0032 
0.0163 
1.9901 
1.002 
Q3:2012 
1.0012 
0.0143 
1.9881 
1.022 
Q4:2012 
1.0047 
0.0178 
1.9916 
1.009 
Q1:2013 
1.0035 
0.0166 
1.9904 
1.003 
Q2:2013 
1.0029 
0.0160 
1.9898 
1.003 
Q3:2013 
1.0031 
0.0162 
1.9900 
0.988 
Q4:2013 
1.0032 
0.0163 
1.9901 
1.006 
Q3:2014 
1.0034 
0.0165 
1.9903 

Q4:2014 
1.0033 
0.0164 
1.9902 

Q3:2014 
1.002 
0.0151 
1.9889 

Q4:2014 
1.0021 
0.0152 
1.9890 

Q3:2015 
1.002 
0.0151 
1.9889 

Q4:2015 
1.0013 
0.0144 
1.9882 

Q3:2015 
1.0012 
0.0143 
1.9881 

Q4:2015 
1.001 
0.0141 
1.9879 

Source: authors’ computations
As we can see in the table above, the inferior and superior limits of the bootstrap intrvals have ranges with low variations. The results are close of the desired monetary policy in Romania, but the intervals are too narrow and the registered inflation rate for inflation is located out of these intervals. The reasons for this fact are related to the underestimated point forecasts for inflation based on linear regression model. All the forecast intervala based on percentilet method include the actual values of inflation rate.
Table 3. Point forecasts and bootstraped forecast intervals using the linear regression model for unemployment rate (%) (percentilet method) (horizon: 2011:Q12015:Q4)
Quarter 
Point forecasts 
Forecast intervals for inflation rate 
Actual values 



Intervals limits 


Q1:2011 
7.21 
5.46 
8.95 
7.20 
Q2:2011 
7.27 
5.52 
9.01 
7.40 
Q3:2011 
7.41 
5.66 
9.15 
7.40 
Q4:2011 
7.41 
5.66 
9.15 
7.40 
Q1:2012 
7.37 
5.63 
9.12 
7.30 
Q2:2012 
7.21 
5.47 
8.96 
7.00 
Q3:2012 
7.04 
5.29 
8.78 
7.10 
Q4:2012 
7.07 
5.33 
8.82 
7.00 
Q1:2013 
7.07 
5.32 
8.81 
7.20 
Q2:2013 
7.24 
5.49 
8.98 
7.30 
Q3:2013 
7.31 
5.56 
9.05 
7.30 
Q4:2013 
7.31 
5.56 
9.05 
7.30 
Q1:2014 
7.33 
5.59 
9.07 
7.20 
Q2:2014 
7.4 
5.66 
9.14 
7.20 
Q3:2014 
7.41 
5.67 
9.15 

Q4:2014 
7.43 
5.69 
9.17 

Q1:2015 
7.45 
5.71 
9.19 

Q2:2015 
7.45 
5.71 
9.19 

Q3:2015 
7.5 
5.76 
9.24 

Q4:2015 
7.53 
5.79 
9.27 

Source: authors’ computations
Starting with 2013, the unemployment rate has a slow tendency of increase. The variations of range for forecast intervals for unemployment rate are rather small, because the differencies between predicted unemployment are low from a quarter to another. All the forecast intervals based on percentilet method include the actual values of unemployment rate.
Table 4. Point forecasts and forecast intervals for qurterly inflation rate and unemployment rate (%) based on historical error methods (horizon: 2011:Q12015:Q4)
Quarter 
Forecast intervals of inflation rate based on historical RMSE of the previous 4 quarters 
Forecast intervals of unemployment rate based on historical RMSE of the previous 4 quarters 


Intervals limits 
Intervals limits 

Q1:2011 
9.448 
11.471 
6.86 
7.55 
Q2:2011 
7.655 
9.673 
6.93 
7.60 
Q3:2011 
4.130 
6.142 
7.03 
7.78 
Q4:2011 
3.121 
5.136 
6.92 
7.89 
Q1:2012 
4.407 
6.415 
6.83 
7.91 
Q2:2012 
6.057 
8.063 
6.65 
7.77 
Q3:2012 
7.731 
9.734 
6.46 
7.61 
Q4:2012 
8.365 
10.375 
6.58 
7.57 
Q1:2013 
7.963 
9.970 
6.58 
7.55 
Q2:2013 
7.799 
9.805 
6.79 
7.68 
Q3:2013 
6.551 
8.557 
6.91 
7.70 
Q4:2013 
4.687 
6.694 
6.95 
7.66 
Q1:2014 
2.917 
4.923 
7.10 
7.56 
Q2:2014 
2.917 
4.923 
7.09 
7.71 
Q3:2014 
2.918 
4.922 
7.03 
7.79 
Q4:2014 
2.919 
4.920 
7.00 
7.86 
Q1:2015 
2.921 
4.917 
6.96 
7.94 
Q2:2015 
2.923 
4.915 
7.05 
7.67 
Q3:2015 
2.928 
4.912 
7.34 
7.87 
Q4:2015 
2.929 
4.911 
7.56 
7.96 
Source: authors’ computations
Forecasts of inflation and unemployment rate provided by this method seem reasonable,the lenght of intervals being rather big. However, if we go in time, these intervals become narrower. All the forecast intervala based on historical error method include the actual values of inflation and unemployment rate.
Considering constant the error from previous year, we will build forecast intervals for inflation and unemployment rate based on two experts’ predictions on the horizon 20042015. Some point forecasts are provided by (Dobrescu, 2013, p. 10).
Table 5. Prediction intervals for annual inflation rate (%) based on historical errors method (horizon: 20042015)
Year 
Forecast intervals based on first expert forecasts 
Forecast intervals based on second expert predictions 
Actual inflation rate 

2004 
3.99 
18.97 
5.24 
18.56 
15.3 
2005 
10.13 
17.35 
3.32 
14.68 
11.9 
2006 
7.82 
9.38 
3.08 
10.92 
9 
2007 
3.90 
7.42 
9.4 
8.06 
6.56 
2008 
1.33 
15.67 
6.03 
11.7 
4.84 
2009 
1.19 
10.01 
7.93 
11.07 
7.85 
2010 
4.81 
7.99 
5.00 
7.40 
5.59 
2011 
3.17 
7.04 
8.29 
9.931 
6.09 
2012 
2.15 
6.85 
5.37 
10.263 
3.3 
2013 
3.19 
12.93 
4.58 
2.13 
3.98 
2014 
3.194 
4.806 
07.18 
2.10 

2015 
3.628 
5.638 
8.18 
2.2201 

Source: authors’ computations
The intervals range for inflation rate is extremly variable in the period 20042012. The range is larger during 20132015. All the forecast intervals based on first expert anticipations include the actual values of inflation rate while only 5 out of 10 intervals on the horizon 20042013 contain the second expert prognosis.
Table 6.Forecast intervals for annual unemployment rate (%) based on historical errors method (horizon: 20042015)
Year 
Forecast intervals based on first expert forecasts 
Forecast intervals based on second expert predictions 
Actual unemployment rate 

2004 
6.808 
7.592 
6.8240 
9.1760 
7.4 
2005 
4.754 
11.066 
4.7640 
11.0360 
6.3 
2006 
4.748 
9.452 
4.0760 
11.5240 
5.9 
2007 
1.638 
11.282 
0.5440 
14.6560 
4 
2008 
3.536 
7.064 
1.5200 
13.2800 
4.4 
2009 
3.400 
13.200 
3.3040 
13.4960 
5.8 
2010 
6.636 
10.164 
7.2040 
7.5960 
7.5 
2011 
6.604 
7.812 
6.3240 
8.6760 
6.9 
2012 
4.748 
9.452 
4.2680 
10.9320 
5.9 
2013 
3.136 
6.664 
3.4320 
6.5680 
7.3 
2014 
5.836 
9.364 
5.4320 
8.5680 

2015 
5.945 
9.567 
5.4734 
8.5834 

Source: authors’ computations
In 2007 the highest range for prediction intervals was obtained for both experts. 9 out of 10 forecast intervals based on first expert anticipations and the second one predictions include the actual values of inflation rate during 20042013. For the last year in the horizon both forecasters anticipated lower unemployment rates.
4. Conclusion
The forecast intervals are a way to reflect the uncertainty that affects the forecasting process. For inflation rate and unemployment rate point predictions forecast intervals were built for Romania, providing a better framework for establishing the decision making process. The annual inflation rate forecasts of the first expert anticipation generated precise prediction intervals when bootstrapping and historical errors methods are applied during 20042013. However, the intervals are quite large. A future direction of research would be the construction of forecast intervals using Bayesian method.
5. Acknowledgement
This article is a result of the project POSDRU/159/1.5/S/137926, Routes of academic excellence in doctoral and postdoctoral research, being cofunded by the European Social Fund through The Sectorial Operational Programme for Human Resources Development 20072013, coordinated by The Romanian Academy.
6. References
Alonso, M., Pena, D. & Romo, J. (2000). Sieve Bootstrap Prediction Intervals. Proceedings in Computational Statistics 14th Symposium, pp. 181186, Utrecht.
Bratu, M. (2012). Forecast Intervals for Inflation in Romania. Timisoara Journal of Economics, 5(1 (17)), pp. 145152.
Croitoru, L. (2013). What Good is Higher Inflation? To Avoid or Escape the Liquidity Trap. Romanian Journal of Economic Forecast, Vol. 16, No. 3, pp. 525.
Dobrescu, E. (2013). Updating the Romanian Economic Macromodel. Journal for Economic Forecasting, Vol. 16, No. 4, pp. 531.
Gospodinov, N. (2002). Median unbiased forecasts for highly persistent autoregressive processes. Journal of Econometrics, Vol. 111, No. 1, pp. 85101.
Guan, W. (2003). From the help desk: bootstrapped standard errors. The Stata Journal, Vol. 3, No. 1, pp. 71–80.
APPENDIX 1. Linear regression model for quarterly index of consumer prices
Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 
C 
0.119341 
0.008761 
13.62264 
0.0000 
Curs_schimb__SA 
0.026202 
0.002331 
11.24136 
0.0000 
Rsquared 
0.700613 
Mean dependent var 
0.022474 

Adjusted Rsquared 
0.695068 
S.D. dependent var 
0.021395 

S.E. of regression 
0.011814 
Akaike info criterion 
6.003922 

Sum squared resid 
0.007537 
Schwarz criterion 
5.931588 

Log likelihood 
170.1098 
Fstatistic 
126.3683 

DurbinWatson stat 
1.032398 
Prob(Fstatistic) 
0.000000 
White Heteroskedasticity Test: 

Fstatistic 
1.284795 
Probability 
0.285184 
Obs*Rsquared 
2.589492 
Probability 
0.273967 
BreuschGodfrey Serial Correlation LM Test: 

Fstatistic 
6.08290 
Probability 
0.191 
Obs*Rsquared 
3.03713 
Probability 
0.305 
APPENDIX 2. Autoregressive model for quarterly unemployment rate
Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 
C 
0.005242 
0.042195 
0.124224 
0.9016 
U(1) 
0.309934 
0.131379 
2.359076 
0.0221 
Rsquared 
0.096677 
Mean dependent var 
0.009259 

Adjusted Rsquared 
0.079305 
S.D. dependent var 
0.322881 

S.E. of regression 
0.309814 
Akaike info criterion 
0.530642 

Sum squared resid 
4.991194 
Schwarz criterion 
0.604308 

Log likelihood 
12.32734 
Fstatistic 
5.565238 

DurbinWatson stat 
1.894308 
Prob(Fstatistic) 
0.022112 
BreuschGodfrey Serial Correlation LM Test: 

Fstatistic 
1.422747 
Probability 
0.238472 
Obs*Rsquared 
1.465554 
Probability 
0.226049 
White Heteroskedasticity Test: 

Fstatistic 
0.352616 
Probability 
0.704547 
Obs*Rsquared 
0.736532 
Probability 
0.691933 
1 PhD, Researcher, Romanian Academy, Institute for Economic Forecasting, Romania, Address: 13, Calea 13 Septembrie, District 5, 76117 Bucharest, Romania, Corresponding author: mihaela_mb1@yahoo.com.
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