Output gap estimates are constructed for India using unobserved components model (UCM) approach on the lines of Watson (1986) and Kuttner (1994). Results from UCMs are not found to be any less sensitive to data revisions when compared to those from the Hodrick-Prescott filter. This, however, could be because of lack of sufficient ‘revised-data’ on which the sensitivity of the results can be tested. Based on standard deviation of change in potential output to data revisions and its ‘economic’ content, the UCM using trimmed mean as the numeraire for inflation comes forth as the best choice. Alternative estimates of “core” inflation, included as a state variable in one of the UCMs, are also provided

Virmani, Vineet

English

Research Papers in Economics

Estimating Output Gap for the Indian Economy
Comparing Results from Unobserved-Components Models and the Hodrick-Prescott Filter
Output gap estimates are constructed for India using unobserved components model (UCM) approach
on the lines of Watson (1986) and Kuttner (1994). Results from UCMs are not found to be any less
sensitive to data revisions when compared to those from the Hodrick-Prescott filter. This, however,
could be because of lack of sufficient ‘revised-data’ on which the sensitivity of the results can be tested.
Based on standard deviation of change in potential output to data revisions and its ‘economic’ content,
the UCM using trimmed mean as the numeraire for inflation comes forth as the best choice. Alternative
estimates of “core” inflation, included as a state variable in one of the UCMs, are also provided
I. Introduction
Importance of a potential output series for analysing macroeconomic phenomena cannot be
overemphasized. It not only enables policy evaluation studies (e.g. analysis of Taylor-type
rules for monetary policy), it also helps in ‘what if’ analysis in both structural and reduced
form models (e.g. in VARs for monetary policy analysis, modeling inflation using structural
models). Also many phenomena are much better understood with output taken as deviation
from a long-run trend (e.g. Phillips curve trade off studies).
Unlike in developed countries like Canada, England, and the US, as of now no official output
gap series exist for India. Rao, Fernandes and Deshpande (1990) earlier estimated potential
output for India, but no attempts were made to extend the series beyond ‘90s. In this study,
taking output series constructed by Virmani and Kapoor (2003), an unobserved component
model (UCM) approach is used to create a potential output series for India for the period
1983Q1 – 2001Q4.
The plan of the paper is as follows. After a brief literature review in section II, unobserved
components models (UCMs) as used in the study are specified in section III. Results are
presented in section IV. Sensitivity analysis of the estimated trend using the three UCMs to
data revisions and comparisons with the Hodrick and Prescott (1980, hereafter HP) and the
modified HPA filter is done in section V. Section VI concludes.
II. Estimating Potential Output
Ideally one would like to have a series for potential output which truly captures the steady
state level of the economy corresponding to the long run aggregate supply curve. It is not
surprising, however, that this approach is not in vogue. Not only are the data requirements
stupendous, the size of a structural econometric model required for such a study, lags
associated with the measurement of the variables (not to mention the noise and the data
revisions problems) makes it both unwieldy and impracticable.
For reasons of speed and ease of estimation time-series based methods have gained
popularity, most popular being the HP filter and the approximate band pass filter of Baxter
and King (1995, hereafter BP). Taken together, Kuttner (1994), Amant and van Norden
(1997), Kichian (1999), Domenech and Gomez (2003) and Rennison (2003) provide a
comprehensive survey of techniques used in literature for estimating potential output.The problem in using filters of the likes of HP and BP is that they are purely empirical in
nature and are essentially ad hoc solution to the problem of trend estimation. If only
estimation of a long-run trend was the concern, the time series based techniques of HP-based
filters provide quite quick and reliable estimates
1. However, as Kuttner (1994) argues, “main
drawback to all these is the lack of substantive economic content.
2” He uses a latent variables
approach to model the unobserved potential output.
Watson (1986) and Clark (1987) were amongst the first to use UCM approach to estimate
potential output. Kuttner (1994) extended the idea and specified “potential as the level of
output at which inflation is constant.
3” Thus, by exploiting a backward looking Phillips curve,
Kuttner (1994) explicitly modeled inflation as a function of the output gap, thereby giving an
economic interpretation to the measure thus constructed.
Furthering the idea of Kuttner (1994), Domenech and Gomez (2003) include “core” inflation,
the NAIRU, and the structural investment rate also in their state space formulation (hereafter
SSF). Thus, using an extended UCM, they are able to extract information about cyclical
output from unemployment and investment series also, thereby adding to the economic
content of the model.
In this study two different UCMs are used on the lines of Watson (1986) and Clark (1987) and
Kuttner (1994) and Domenech and Gomez (2003). Details follow in the next section.
III. Model Specification
¾  Output: Following Watson (1986), output is separated into a trend and a cycle. The trend
component is assumed to follow a random walk with drift and the cyclical component is
assumed to follow an AR (2) process (much popular with the real business cyclical
theorists; see Romer, 1996, Ch. 4). Thus, (natural logarithm of) output is specified as:
*
1
**
1
11 2 2
2
2
~( 0 ,)
~( 0 ,)
tt t
tt t
tt t t
t
t
yy z
yy
zz z
N
N
ε
η
αε
ϕϕη
εσ
ησ
−
−
−−
=+
=+ +
=++
[1]
                                                
1 The estimates are still sensitive to the end-of-the-sample problems
2 K. Kuttner (1994), “Estimating Potential Output as a Latent Variable,” Journal of Business and Economic
Statistics, 12, 3, p. 362
3 ibid, p. 364¾  Inflation: As found by Kuttner (1994) for the U.S., a parsimonious backward looking
Phillips curve specification with MA(2) errors fits well for inflation in India too
4:
*
11 1 2 2
**
1
2
2
~( 0 ,)
~( 0 ,)
tt t t t t
tt t
t
t
z
N
N
ν
ς
ππβ νδ ν δ ν
ππ ς
νσ
ςσ
−− −
−
=+ + + +
=+
[2]
where, following Domenech and Gomez (2003), core inflation (
*
t π ) is modeled as a random
walk without drift.
Note how ‘restriction’ on the coefficient of core inflation as above allows for its interpretation
as that level of inflation when the output gap, 
1 t z
−  is zero. If in first equation in [2], 
1 t z
−  is 0,
with E(
t ν ) = 0, it follows that E(
t π ) = 
*
t π .
Equations [1] and [2] can be conveniently cast as a State Space Model (SSM), facilitating
estimation of the latent variables by Maximum Likelihood (ML) using Kalman Filter. Details
can be found in Harvey (1993). For above specification, the SSM is:
-1   tt t
tt
cA
yH
αα ξ
α
=+ +
=
[3]
with the state vectors and system matrices in the two models given as below:
¾  UCM-1:  On the lines of Watson (1986) and Clark (1987)
[]
*
t
t
t t
t-1
t-2
a1 0 0 0 y
000 z
=c =      A  = = H  = 1 0 0 0
00 1 0 0 0 z
00 0 1 0 0 z
τ
τ
ε
ϕϕ η
αξ
12
     
     
     
     
     
      
                                                
4 Other specification for inflation were also checked; MA(2) was selected using the general to specific criterion¾  UCM-2 and UCM-3: On the lines of Kuttner (1994) and Domenech and Gomez (2003),
with inflation alternatively based on WPI-All Commodities Index and a Trimmed Mean
*
t t
12 t t
t-1
t-2
t t *
t
t
t-1
t-2
a 10 0 00000 y
0 0 00000 z
0 01 0 00000 0 z
0 00 1 00000 0 z
=c =      A  = =
0 00 0 01000
0 00 0 00000
0 00 0 00100
0 00 0 00010
ε
ϕϕ η
αξ
π
ν
ν
ν
   
   
   
   
   
   
    ς
   
   
   
   
       
t
t
12
0
0
110 0000 0
H =
00 011
ν
βδ δ











 



IV. Estimation and Results
After running the Kalman Filter recursions as given in Harvey (1993), the state vector along
with their associated Mean Squared Errors (MSEs) and the hyperparameters can be estimated
using ML. The likelihood function is proportional to:
'1
1
||
n
tt t
t
L( ) -ln F e F e θ
−
=
=− ∑ [4]
where θ  is the vector of the hyperparameters, and F is the MSE associated with error, e.
To estimate the vector of hyperparameters, we minimize the negative of the likelihood
functionL( ) θ   using the Nelder-Mead simplex search method available in MATLAB
5.
Although Nelder-Mead is one of the slower search routines, it is more reliable provided the
initial values are not too off-mark, which is not a concern for the problem at hand.
Data
For output, quarterly estimates of GDP at factor cost (1993-94 = 100) constructed by Virmani
and Kapoor (2003) have been used after seasonally adjustment by the TRAMO/SEATS
6
method
7. Inflation is alternatively taken to be based on seasonally adjusted WPI-All
Commodities (1993-94 = 100) and 49/50 Trimmed Mean
8.
                                                
5 Using the function fminsearch available in the Optimization Toolbox of MATLAB 6.5
6 Time Series Regressions with ARIMA Noise/Signal Extraction in ARIMA Time Series
7 Using the software DEMETRA made available by the European Statistical Institute (EUROSTAT)
8 See Virmani (2003) for selection of the optimal trimming patternInitialization of the Hyperparameters
Running the HP filter on the output and the inflation series, and estimating OLS for models in
[1] and [2] gives initial estimates of the hyperparameters. Results are reported in Table 1
below.
Table 1
Initialization of the Hyperparameters
Hyperparameter
1 ϕ
2 ϕ a β
1 δ
2 δ
2
ε σ
2
η σ
2
ς σ
2
ν σ
Initial Value 0.54 0.18 -0.0094 -0.2 -0.7 -0.2 0.0000012 0.00011 0.0000018 0.002
Initialization of the State Vector
Since both potential output and core inflation have been modeled as nonstationary, unlike for
a stationary state space model, initial conditions for the Kalman Filter are not well defined.
However, since we have first estimates for potential output from running the HP filter, and
that of output gap from the OLS estimates, we can treat the initial condition as ‘known’ for
our purpose. Taking first three values from the HP filtered output series, cyclical output is
initialized as the residual, 
*
tt yy − . For the MA terms corresponding to inflation their
expectation (zero) is used to for initialization. MSE of the initial state vector (taken to be
diagonal) are taken from OLS estimates from [1] and [2]. Since inflation and cyclical output
have been modeled as MA(2) and AR(2) process respectively, essentially filtering starts from
the fourth observation. Initial values are reported in Table 2 below.
Table 2
Initialization of the State Vector
State Variable
*
t y t z
t-1 z
t-2 z
*
t π t ν
t-1 ν
t-2 ν
Initial State Value  (
0 α ) 11.67 0.0056 0.0032 0.0093 0.061 0 0 0
Initial State MSE  (
0 P ) 0.096 0.0002 0.0002 0.0002 0.0003 0.0002 0.0002 0.0002
Results and Discussion
Results from the three UCMs are reported in Table 3 below
9. Filtered and Smoothed series
from the three models are plotted in Figure 1, along with comparisons with results from the
HP filter and the modified HPA filter. HPA is HP filter on extended series using a suitably
selected ARIMA model. Kaiser and Maravall (2000) show using Monte Carlo experiments
that HPA is less sensitive to end of sample observations. For output data used in the study, an
IMA(1) was found suitable. “Core” inflation from UCM-2 and UCM-3 are compared against
estimates from the HP filter and the 49/50 Trimmed Mean in the last quadrant of Figure 1.
Results from all the models are broadly in agreement, especially at the turning points.
Though, there is significant divergence at the end of the sample when results are compared
with HPA. As would be expected, when trimmed mean is used as the numeraire for inflation,
                                                
9 For smoothing Fixed-interval algorithm was usedestimates of output gap are quantitatively smaller. From the last quadrant in Figure 1, a
striking feature is ‘over-estimation’ of “core” inflation when WPI-All Commodities is used as
the measure for inflation (in UCM-2), suggesting that high noise in the inflation series (as
shown by Virmani (2003) in a detailed statistical analysis of the components underlying the
WPI-All Commodities index) could have possibly distorted estimates of long-run trend.
Table 3
Maximum Likelihood Estimates of the Hyperparameters
Hyperparameter
1 ϕ
2 ϕ a β
1 δ
2 δ
2
ε σ
2
η σ
2
ς σ
2
ν σ
UCM – 1 0.66 0.21 0.014 - - - 0.0000009 0.00013 - -
UCM – 2 0.57 0.34 0.014 -0.39 0.19 -0.24 0.0000022 0.00013 0.000002 0.0022
UCM – 3 0.63 0.27 0.014 -0.11 0.19 0.22 0.000014 0.00011 0.000006 0.00011
Figure 1
V. Sensitivity Analysis and Out-of-Sample Performance of the UCMs
Sensitivity Analysis to Data Revisions
It is well known that contemporaneous national income data undergoes various revisions
before finalization. Orphanides and van-Norden (1999) among others have shown that policy
suggestions using Taylor-type rules are highly sensitive to vintage of the data used. Thus, it is
important to have a measure for potential output that is not very sensitive to data revisions. In
this section robustness of different measures of potential output is tested to end-of-sample
data revisions. In particular, standard deviation of change in potential output series iscalculated for the three UCMs, the HP filter and the modified HPA filter. However, this test
on Indian data can at best be illustrative, because it has only been four years since CSO has
started releasing quarterly data
10, thereby limiting the ‘number’ of revised data to only four.
Out of that, it was noticed that first release of quarterly estimates were revised extraordinarily.
Disregarding first two revisions, this leaves us with only the penultimate year on which
sensitivity of results to data revisions can be tested. This is a problem because ideally one
would want to see revisions over a sufficient length of time to be able to notice the sensitivity
of the potential output series to data revisions.
Table 4 below lists the standard deviation in the changes in the series when compared with
data of the penultimate vintage. Smoothed estimates from UCMs have been used for
comparison. Potential output series estimated using the two vintage of data for all the models
are presented in Figure 2. Estimates from UCM-2 look to be most sensitive, and HP/HPA
least, but as argued above, these comparisons are at best illustrative. For the period prior to
1996 only a single estimate exist, and not the quick, advanced and revised estimates of
quarterly output which now CSO makes available since 1999 .
Table 4
Comparison of Standard Deviation in Changes in Potential Output on Data Revisions
Method HP HPA UCM-1 UCM-2 UCM-3
Standard Deviation 0.0039 0.0026 0.0019 0.0091 0.0047
Figure 2
                                                
10 First appearing in National Account Statistics, 1999Out-of-sample Performance of the UCMs
For the out-of-sample performance of the UCMs, ‘smoothed’ predictions of output and
inflation from the UCMs are compared against actual. In-sample comparison is presented in
Figure 3. Clearly, the Kalman Filter does a good job of using the information contained in the
sample.
Figure 3
For out-of-sample performance of the UCMs, the Kalman Filter recursions are run to get the
predicted values and the associated prediction MSE. Using the MSEs, density forecasts of
quarter-to-quarter percentage growth in output for subsequent eight quarters (starting
2002Q1) are plotted in the top half of Figure 4 below
11.  The density shown covers roughly
95% percent of the probability distribution (with increments of 0.5σ till ±2.5σ)
12 – with
‘darker’ bands indicating region of higher probability. Since as of now only one out-of-
sample observation exists for output (CSO has only recently released ‘first’ estimates for
GDP at factor cost for 2002Q1. From Virmani (2003), however, data on WPI at 1993-94
prices and the associated trimmed mean are available for four subsequent quarters, which are
plotted as ‘circles’ in the lower half of Figure 4. Though not of much relevance in the context
of measurement of potential output, such ‘fan charts’ are much in vogue with central banks
across the world
13 to communicate their view of future inflation and output.
                                                
11 Performance of UCM-2 and UCM-3 was found to be quite similar, hence only one is shown
12 here σ is square root of the MSE of the predicted values
13 Such releases have become official statements with Sveriges Riksbank, Sweden and the Bank of England, U.KVI. Conclusion
Estimates of potential output/output gap for India have been provided using unobserved
components model approach. Broadly results are in agreement. Though results from UCM-1
and HPA are quantitatively superior, since UCM-3 uses more information and is void of noise
in high frequency inflation data, its use is recommended to estimate output gap. For a
thorough validation of UCM-3, however, we must wait till we have more releases of data
from the CSO.
Figure 4References
Baxter, M. and R. King (1999) “Measuring Business Cycles: Approximate Band-Pass Filters
for Economic Time Series.” Review of Economics and Statistics 81(4): 575–93.
Clark, P.K (1987) “The Cyclical Component of U.S. Economic Activity,” Quarterly Journal
of Economics 102(4): 797–814.
de Jong, P. (1991) “The Diffuse Kalman Filter,” Annals of Statistics, 19, 1073-1083.
Domenech, R. and V. Gomez (2003) “Estimating Potential Output, Core Inflation and the
NAIRU as Latent Variables,” mimeo University of Valencia
Harvey, A. (1993) Time Series Models, 2
nd Edition, Harvester-Wheatsheaf, London
Hodrick, R.J. and E.C. Prescott (1980) ‘‘Post-War U.S. Business Cycle: An Empirical
Investigation’’ Center for Mathematical Studies in Economics and Management
Science, Northwestern University Discussion Paper No. 451.
Kaiser, R. and A. Maravall (2000) “Estimation of the Business Cycle: A Modified Hodrick
Prescott Filter” Spanish Economic Review, 1, 175-206
Kichian, M. (1999) ‘‘Measuring Potential Output within a State-Space Framework,’’ Bank of
Canada WP No. 99–9
Kuttner, K.N. (1994) ‘‘Estimating Potential Output as a Latent Variable,’’ Journal of
Business and Economic Statistics 12(3): 361–68.
Orphanides, A. and S. van Norden (1999) “The Reliability of Output Gap Estimates in Real
Time,” Board of Governors of the Federal Reserve System Finance and Economics
Discussion Series No. 38
Rao, M. J. M, C. J. Fernandes, and C. S. Deshpande (1990) “Potential Rates of Growth for the
Indian Economy: Recursive Estimation Results,” Journal of Indian School of Political
Economy 2, 110-138
Rennison, A. (2003) “Comparing Alternative Output-Gap Estimators: A Monte Carlo
Approach,” Bank of Canada WP No. 03–8
St-Amant, P. and S. van Norden (1997) “Measurement of the Output Gap: A Discussion of
Recent Research at the Bank of Canada,” Bank of Canada Technical Report No. 79
Virmani, V. (2003), “Distribution of Cross-section of Price Changes and Measurement of
Inflation,” Indian Institute of Management, Ahmedabad WP 2003-12-03
Virmani, V. and R. Kapoor (2003), “Developing a Back Series of Monthly and Quarterly
Income Estimates for India: 1983Q1 – 1999Q4,” Indian Institute of Management, Ahmedabad
WP 2003-10-02
Watson, M.W. (1986). “Univariate Detrending Methods with Stochastic Trends,” Journal of
Monetary Economics 18: 49–75

Estimating Output Gap for the Indian Economy: Comparing Results from Unobserved-Components Models and the Hodrick-Prescott Filter

http://www.iimahd.ernet.in/assets/snippets/workingpaperpdf/2004-04-02vineet.pdf

Estimating Output Gap for the Indian Economy: Comparing Results from Unobserved-Components Models and the Hodrick-Prescott Filter

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics