An analysis of CBI’s in‡ation forecast errors
A report to taskforce on reviewing Iceland’s monetaryand currency policies
Marías Halldór Gestsson
May 2018

1Introduction

The Central Bank of Iceland (CBI) started using in‡ation targeting as a framework for monetary policy in March 2001. Following that, it started publishingin‡ation forecasts on a quarterly basis in the Monetary Bulletin. In this paper,
we present an anlysis of the errors of these forecasts.
Uncertainty is natural in forecasting since forecasts are based on estimatedeconomic models containing uncertain shocks as well as uncertainty due to estimated parameters etc. Hence, forecast errors are inevitable and are not due tobad modelling or bad forecasting. However, one requires that forecasts are notsystematically wrong (unbiasedness) and make use of all available information
at the time of a forecast (e¢ ciency). Further, it is required that forecasts madeusing sophisticated economic model, like the QMM model (see Daníelsson, et.
al., 2015) used by the CBI, do better in forecasting than simple naive models(accuracy).
The quality of in‡ation forecasts (accuracy, unbiasedness and e¢ ciency) are
important from the viewpoint of implementing in‡ation targeting since in‡ation
targeting entails in‡ation forecast targeting (see Svensson, 1997), i.e. central
banks use in‡ation forecasts when deciding on their policy rates under in‡ation
targeting. The quality of in‡ation forecasts is also important from a welfare
viewpoint since these are important tools for central banks in anchoring in‡ation
expectations, i.e. the less the quality of in‡ation forecasts the less credible
they are and the more di¢ cult it is for central banks to get economic agents(individuals, …rms etc.) to form their in‡ation expectations based on them.
The paper is organized as follows: The data used is discussed in chapter
2. Chapter 3 gives an overview of the CBI’s in‡ation forecast errors in 20012017 and chapter 4 gives an analysis of performance or quality of the in‡ation
forecasts. Chapter 5 concludes the paper.

12

Data

In‡ation in a quarter is here de…ned and measured as an (relative) increasein the CPI from the same quarter last year and the data frequency is therefore
quarterly. Data on CPI was obtained from the CBI’s QMM database (see QMMdatabase) and data on in‡ation forecasts was obtained from the bank’s quarterly
Monetary Bulletin in 2001 - 2017 (see Monetary Bulletin).
As is discussed above, the CBI started using in‡ation targeting in March
2001. We therefore analyse its in‡ation forecasts since that date in this paper.
For most of the quarters in the period, the bank has published in‡ation forecasts
for up to eight to twelve quarters ahead. An overview of the forecasts madeduring the period 2001-2017 is given in the following table:Table 1. CBI’s in‡ation forecasts in 2001-2017
published in the Monetary Bulletin
Forecasts made inQuarters ahead
2001Q20 10
2001Q30 9
2001Q4-2004Q2, 2004Q4-2006Q2, 2006Q40 8
2007Q1-2007Q2, 2007Q4, 2008Q2-2009Q30 11
2009Q4-2017Q30 12

where 2001Q2 stands for the second quarter of 2001 etc. and 0 quarters
ahead is the quarter in which the forecast is made. For most of the forecasts,
the dates in the table correspond to the dates of publications of the Monetary
Bulletins.13In‡ation forecast errors

In‡ation forecast error for a forecast q quarters ahead made in quarter t isde…ned as:
et+q;t(1)t+q;tt+q
where q 0, t+q;t is in‡ation forecast q quarters ahead made in quarter t andt+q is realized in‡ation in quarter t + q. Hence, a positive in‡ation forecast
error means that forecasted in‡ation is higher than realized in‡ation etc.
The following …gure shows forecast errors for in‡ation foracests one, fourand seven quarters ahead:2
1 There are few exceptions to this. For example, the forecast published in the Monetary
Bulletin 2006/2 dated July 1st 2006 (third quarter of 2006) is dated 2006Q2 (instead of2006Q3) in this paper since data on in‡ation for the second quarter of 2006 was not available
on July 1st 2006.
2 Forecast errors for forecasts 0-12 quarters ahead are shown in the appendix.

2

where the dates on the x-axis are the quarters for which forecasts are made(and not the quarters in which foracasts are made). For example, the value ofthe black line in the fourth quarter of 2004 is the in‡ation forecast error for thatquarter of a forecast made in the third quarter of 2004. The breaks in the lines
are due to missing forecasts (see table 1).
Figure 1 (and the …gures in the appendix) shows extreme forecast errors in2008 and 2009, which are due to the global …nancial crises and the boom-bust
cycle of the Icelandic economy during these year. Further, the …gure shows thatin‡ation forecast errors have been positive during the past few years implyingthat the CBI’s in‡ation forecasts for the quarters of these years have been higher
than realized in‡ation.
Including forecasts made for the quarters of 2008 and 2009 in the analysis
below would have large impact on the results and preclude interpretation of theresults as representing "normal times". We therefore exclude forecast errors forthese years in the analysis below. Summary statistics for the data on forecast
errors and p-values for normality tests are given in the following table:

3

Table 2. Summary statistics and normality tests
q Av. (%) St. dev. (%) p-val.3 Obs.
00:09
0:31
0:18
55
1
0:16
0:97
0:15
54
20:03
1:43
0:39
54
30:28
1:96
0:10
53
40:30
2:39
0:07
52
50:39
2:34
0:35
51
60:48
2:00
0:11
51
70:61
1:76
0:18
50
80:69
1:70
0:33
48
90:53
1:85
0:06
32
100:491:70
0:08
30
110:52
1:69
0:09
29
120:26
1:02
0:03
20
One quarter ahead forecasts were on average 0:16% higher than realized
in‡ation the period 2001-2017 (excluding 2008 and 2009) and the standard deviation of one quarter ahead forecast errors was 0:97%. In‡ation foreasts werehigher than realized in‡ation on average for 0 2 quarters ahead forecasts but
lower than realized in‡ation for the longer forecasts (more than two quartersahead forecasts) except for twelve quarters ahead forecasts.
It is expected that the variance, or the standard deviation, of forecast errors
increases with the number of quarters ahead forecasted since there is more uncertainty the longer the forecast horizon. Here, this only holds for 0 4 quarters
ahead forecast errors. It should be noted that the CBI did not start publishing
9 12 quarters ahead forecasts on a regular basis until the end of 2009, which
results in fewer observations (obs.) for these forecasts. This may at least partly
explain this result.
The null hypothesis of normally distributed forecast errors cannot be rejected for forecasts 0 2 and 5 8 quarters ahead while the results of thenormality test for the errors of 3, 4 and 9 11 quarters ahead forecasts depend on the signi…cance level chosen (5% or 10%). The null hypothesis can berejected for forecasts 12 quarters ahead. These results are important for theanalysis of unbiasedness and e¢ ciency below since the sample size (the numberof observations) is limited.

4Forecast performance

We use three critiques for evaluating in‡ation forecast performance of the CBI’sin‡ation forecasts: accuracy, unbiasedness and e¢ ciency. Accuracy gives a measure of how close forecasts were to realized in‡ation during the period 2001-2017,
i.e. the smaller the forecast errors are the greater is e¢ ciency. Unbiasedness
3 Doornik-Hansen test for normally distributed forecast errors.

4

shows whether in‡ation forecasts are systematically too high (biased upward)
or too low (biased downward) or neither (unbiased) and e¢ ciency shows how
well forecasters use the available information when forecasting, such that betteruse of available information gives more e¢ cient forecasts.
Due to missing forecasts prior to 2010 and omitting forecasts for the quartersof 2008 and 2009 from the anlysis, we only use data on forcasts and forecast
errors for forcasts made for the quarters of 2010-2017 in the anlysis below.

4.1

Accuracy

We use two measures of forecast accuracy; mean absolute forecast error (M AE)
and root mean squared forecast errors (RM SE). The …rst gives the mean ofnumerical values of forecast errors:
M AEq =1obs

2017Q3X q

t=2010Q1 q

jet+q;t j(2)over the sample while the second gives the square root of the mean of squared
forecast errors:vuX qu 1 2017Q3t
RM SEq =e2t+q;t(3)obs
t=2010Q1 q

The higher M AE and RM SE are the greater are deviations of forecasted in‡ation from its realized values and the less is forecast accuracy. Note that larger
forecast errors get more weight in RM SE since forecast errors are raised to thesecond power.
M AEq can be interpreted as the average deviation of forecasts q quartersahead from realized in‡ation during the period while RM SEq is non-informativeon its own and should be used in comparison to other forecasts (other models,other countries, di¤erent time periods etc.). Here we compare the RM SEq -sof the CBI’s in‡ation forecast to a naive forecast where in‡ation is assumed tofollow a random walk process and, hence, forecasted in‡ation in current (q = 0)
and future quarters (q = 1; :::; 12) is given by in‡ation in the previous quarter:t+q;t =t 1

The results are given in the following table:

5Table 3. Measured M EAq and RM SEq ratio
qM EAq (%)

RM SEqCBI
RM SEqN

Obs.

01234
56
7
8
9
10
11
12

0:27
0:75
1:07
1:38
1:79
1:81
1:67
1:67
1:58
1:46
1:33
1:28
0:93

0:37
0:61
0:60
0:64
0:67
0:59
0:47
0:43
0:40
0:38
0:34
0:33
0:35

31
31
31
31
31
31
31
31
30
29
28
27
20

The deviation of one quarter ahead forecasts from realized in‡ation was on
average 0:75% during the period and the forecast accuracy is greatest for zeroquarters ahead forecasts (M EA0 = 0:27%). The average deviation is increasing
with the number of quarters ahead forecasted up to …ve quarters and thendecreasing implying that forecast accuracy is …rst decreasing with the numberof quarters ahead forecasted and then increasing. Forecasts …ve quarters ahead
were the least accurate ones during the period.
The ratio between RM SE of CBI’s in‡ation forecasts (RM SE CBI ) andnaive forecasts (RM SE N ) shows that it is less than one for all quarters ahead
forecasted implying that the CBI’s forecasts were more accurate than naiveforecasts during the period.

4.2

Unbiasedness

A forecast is unbiased if the mean of its forecast error equals zero; E (et+q;t ) = 0,indicating that there are no systematic errors in the forecast. However, a forecast
is biased upward (downward) if the mean is positive (negative); E (et+q;t ) > 0(E (et+q;t ) < 0), indicating that a forecast is systematically too high (too low).
An estimate of bias in CBI’s in‡ation forecasts is obtained by estimating thefollowing equation by OLS using data on errors of forecasts made for 2010Q12017Q3 for each of forecasts 0 8 quarters ahead (q = 0; 1; 2; :::; 8):et+q;t =q + ut+q;t(4)where q is a parameter and ut+q;t is a residual with zero mean and a constant variance. We therefore have that E (et+q;t ) = q implying that the mean
in‡ation forecast error of a forecast q quarters ahead is non-zero and in‡ation
forecast therefore biased if q 6= 0. Further, it is biased upward if q > 0 and
downward if q < 0. The estimation results are given in the following table:6Table 4. Results from estimation of equation (4)4qObs.
q (p-value )00:0012 (0:02)
31
10:0036 (0:02)
31
20:0039 (0:16)
31
30:0003 (0:96)
31
40:0008 (0:90)
31
50:0024 (0:71)
31
60:0030 (0:60)
31
70:0039 (0:47)
31
80:0041 (0:43)
30

The null hypothesis of unbiasedness ( q = 0) is rejected for zero and one
quarters ahead forecasts while it cannot be rejected for 2 8 quarters ahead
forecasts. Further, since 0 > 0 and 1 > 0, the data indicates that zero andone quarters ahead forecasts are biased upward indicating that the CBI systematically forecasts too high in‡ation for the current (q = 0) and the followingquarter (q = 1).

4.3

E¢ ciency

A forecast is e¢ cient if it makes use of all available information when it ismade. This implies that the forecast error should be uncorrelated with all
information available to forecasters at the time a forecast is made. One suchpiece of information is last quarter’s in‡ation.
To analyse e¢ ciency we estimate the following equation by OLS for each offorecasts 0 8 quarters ahead (q = 0; 1; 2; :::; 8) using data on forecast errors offorecasts made for 2010Q1-2017Q3 and in‡ation in 2008Q1-2017Q2:
et+q;t =q +q t 1 + ut+q;t(5)where q and q are parameters, t 1 is in‡ation in the quarter before a forecast
is made (known information) and ut+q;t is a residual with zero mean and aconstant variance. q 6= 0 implies non-zero correlation between known in‡ationand in‡ation forecast errors of a forecast q quarters ahead and, hence, thata forecast is not e¢ cient. This implies that forecasters could improve theirforecasts by accounting more for past in‡ation in their forecasts. Further, if
q > 0 ( q < 0) then forecasters predict high (low) in‡ation if it is high in thequarter prior to the one a forecast is made implying that they predicted more(less) persistency in in‡ation than is supported by data. The estimation results
are given in the following table:
4 Two-sided t-test forrelation in residuals.

q = 0 using HAC standard errors to account for potential autocor-7q012345678Table 5. Results from estimation of equation (5)56R2p-val.7 Obs.
q (p-value )q (p-value )0:0005 (0:65)
0:0505 (0:06)
0:10
0:05
31
0:0000 (0:99)
0:0971 (0:06)
0:06
0:30
31
0:0021 (0:68)
0:0453 (0:50)
0:01
0:03
31
0:0128 (0:07)
0:2830 (0:03) 0:25
0:01
31
0:0173 (0:03)
0:3692 (0:00) 0:35
0:24
31
0:0178 (0:02)
0:3842 (0:00) 0:44
0:12
31
0:0144 (0:04)
0:3142 (0:00) 0:42
0:00
31
0:0107 (0:12)
0:2553 (0:01) 0:300:00
31
0:0092 (0:17)
0:2283 (0:01) 0:26
0:03
30

The null hypothesis of e¢ cient forecasts ( q = 0) can be rejected for all
forecasts except those 0 2 quarters ahead, where the results of the tests forforecasts 0 and 1 period ahead depend on the signi…cance level chosen (5% or10%). Further, since 3 , 4 , 5 , 6 , 7 and 8 are all negative, this indicatesthat the CBI predicted too little persistency in in‡ation. Note that the null
hypothesis of normally distributed residuals in (5) can be rejected for all except
forecasts 1, 4 and 5 quarters ahead (q = 1; 4; 5). Due to the small sample used
here, this should be kept in mind when interpreting the results of the signi…cancetest in table 5.

5Conclusions

The main conclusions of the analysis are the following:In‡ation forecasts made by the CBI for the …rst quarter of 2010 to thethird quarter of 2017 were much more accurate than naive ones implyingthat using a sophisticated model to forecast in‡ation resulted in forecasts
being much more accurate.
The data indicates that zero and one quarter ahead forecasts made by
CBI are biased upward indicating that the CBI predicts too high in‡ationin their current and next periods forecasts.
The data indicates that forecasts three to eight quarters ahead are not e¢ cient indicating that the CBI does not make use of all available information
when forecasting. Further, the data indicates that the CBI predicts too
little persistency in in‡ation.
5 Two-sided t-test for
q = 0 using HAC standard errors to account for potential autocorrelation in residuals.
6 Two-sided t-test for
q = 0 using HAC standard errors to account for potential autocorrelation in residuals.
7 Doornik-Hansen test for normally distributed residuals in (5).

8References[1] Daníelsson, Ásgeir, Bjarni G. Einarsson, Magnús F. Guðmundsson, Svava J.
Haraldsdóttir, Thórarinn G. Pétursson, Signý Sigmundardóttir, Jósef Sigurðsson and Rósa Sveinsdóttir (2015), QMM, A Quarterly Macroeconomics
Model of the Icelandic Economy, Version 3.0, Central Bank of Iceland’sWorking Paper No. 71[2] QMMdatabase:https://www.sedlabanki.is/utge…d-efni/frettir-ogtilkynningar/frettasafn/frett/2017/11/15/Peningamal-2017-4/[3] Monetary Bulletin: https://www.cb.is/publications/publications/publicationsall-archive/?all=1&[4] Svensson, Lars (1997), In‡ation forecast targeting: Implementing and monitoring in‡ation targets, European Economic Review, 41, pp. 1111-1146

Appendix

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