Estimating standard error of inflation rate in Pakistan: a stochastic approach.
Iqbal, Javed ; Hanif, M. Nadim
We estimate standard errors (S.Es.) of month on month and year on
year inflation in Pakistan based on data for the period of July 2001 to
June 2010 using the stochastic approach as well as extended stochastic
approach to index numbers. We develop a mechanism to estimate S.E. of
period average headline inflation (rate) using these approaches. This
mechanism is then applied to estimate S.Es. of 12-month average rate of
inflation in Pakistan for July 2003 to June 2010. The systematic changes
in the relative prices of different groups in the CPI basket for
Pakistan are also estimated. The highest (positive) relative price
inflation occurred in 'food, beverages and tobacco' group and
the lowest (negative) for 'recreation and entertainment'
group, during fiscal year (FY) 01 to FY10.
JEL classification: C13, C43, E31
Keywords: Estimation, Index Numbers, Inflation Rate, Standard Error
Estimating Standard Error of Inflation Rate in Pakistan: A
Stochastic Approach
"The answer to the question what is the mean of a given set of
magnitudes cannot in general be found, unless there is given also the
object for the sake of which a mean value is required. There are as many
kinds of average as many purposes; and we may almost say in the matter
of prices as many purposes as many writers." Edgeworth (1888).
1. INTRODUCTION
The stochastic approach to index numbers has recently attracted
renewed attention of researchers as it provides the standard error of
index number (and its growth). One of the most important uses of index
number is in the case of measurement of the general price level in an
economy (and then inflation, of course). This approach has been applied
to measure the rate of inflation (1) in studies like Clement and Izan
(1987), Selvanathan (1989), Crompton (2000), Selvanathan (2003),
Selvanathan and Selvanathan (2004), and Clement and Selvanathan (2007).
Historically, there are two main approaches to measure the index
number: the functional approach and the stochastic approach. In the
functional approach, prices and quantities of various goods and services
are considered as connected by certain typical observable relationship
[Frisch (1936)]. The stochastic approach is less well known although it
has a long history dating back to Jevons (1865) and Edgeworth (1888). In
the stochastic approach prices and quantities are considered as two sets
of independent variables. It assumes that (ideally) individual prices
ought to change in the same proportion from one point of time to the
other. This assumption is based upon the quantity theory of money--as
the quantity of money increases all prices should increase
proportionally. Any deviation of individual prices from such
proportionality is seen as 'errors of observation' and/or may
be the result of non-monetary factors' effect on prices. Thus the
rate of inflation can be calculated by averaging over the proportionate
changes in the prices of all individual goods and services.. Keynes
(1930) criticised the equiproportionate assumption as being 'root
and cause erroneous'. In the functional approach the deviations
from proportionality are taken as expressions for those economic
relations that serve to give economic meaning to index numbers [Frisch
(1936)].
The recent interest of researchers in the stochastic approach to
index number theory is led by Balk (1980), Clements and Izan (1981,
1987), Bryan and Cecchetti (1993) and Selvanathan and Rao (1994).
Clements and Izan (1987) recognised the Keynes (1930) criticism on the
assumption of identical systematic changes in prices and viewed the
underlying (2) rate of inflation as an unknown parameter to be estimated
from the individual price changes by linking the index number theory to
regression analysis.
Again, using the functional approach to index numbers we obtain an
estimate of the inflation rate without knowing its distribution. Thus,
we have no basis to make any statistical comment, say about
'efficiency,' of the estimated inflation rate for which we
shall also need the standard error of the estimated rate. The stochastic
approach leads to familiar index number formulae such as Divisia and
Laspeyres. As uncertainty plays a vital role in this approach, the
foundations differ markedly from those of the functional approach;
linking the index number theory to regression analysis we not only get
an estimate of the rate of inflation but also its sampling variance.
With the relaxation of the assumption that prices of goods and services
change equiproportionately, individual prices in the basket of price
index move disproportionately (which usually happens) and thus the
overall rate of inflation may become less well defined [Selvanathan and
Selvanathan (2006b)]. In such situations the ability of the stochastic
approach becomes important as it allows us to construct confidence
interval around the estimated rate of inflation with the help of
standard error (of inflation). Confidence interval can be used for some
practical purposes such as wage negotiations, wage indexation, inflation
targeting (in interval), etc.
One of the criticisms on this new stochastic approach of Clements
and Izan (1987) is on the restriction of homoscedasticity on the
variance of the error term in the OLS regression [Diewert (1995)].
Crompton (2000) also pointed out this deficiency and extended the new
stochastic approach to derive robust standard errors for the rate of
inflation by relaxing the earlier restriction on the variance of the
error term by considering an unknown form of heteroscedasticity.
Selvanathan (2003) presented some comments and corrections on
Crompton's work. Selvanathan and Selvanathan (2004) showed how
recent developments in the stochastic approach to index number can be
used to model commodity prices in OECD countries. Selvanathan and
Selvanathan (2006) calculated the annual rate of inflation for
Australia, UK and US using the stochastic approach. (3) These studies
provided a mechanism for calculating the standard error for inflation.
Rather than targeting the headline (year on year or YoY) inflation, some
countries track 12 month moving average inflation as the goal of
monetary policy. However, there is no work in the literature to estimate
the standard error of period average inflation. We contribute by
developing a mechanism to estimate the standard error of period average
inflation. (4)
In this study we estimate standard errors of month on month (MoM)
and YoY inflation in Pakistan using the stochastic approach, following
Selvanathan and Selvanathan (2006). Since the State Bank of Pakistan
(the central bank) targets 12-month average of YoY inflation, we
contribute by applying our mechanism to estimate the standard error of
12-month average inflation in Pakistan.
The criticism on the assumption that when prices change they change
equally proportionally, has been responded to by Clements and Izan
(1987) who extend the stochastic approach by considering the underlying
rate of inflation separate from the changes in relative prices. We also
estimate the standard errors of (MoM, and YoY) inflation in Pakistan
using the same approach. We contribute by developing a mechanism, and
applying this to Pakistan, to estimate the standard error of period
average inflation also using the same approach. Furthermore, by applying
this approach we also estimate the systematic (MoM, and YoY, and
12-month average) change in relative prices based upon individual prices
of 374 commodities in the CPI basket of Pakistan for the period July
2001-June 2010. However, in this paper we present only the average
systematic change in relative prices of different groups in the CPI
basket.
In the following section we provide the details of the existing
mechanisms of stochastic approach and their applications in the index
number theory in the context of price index. We then further build upon
this approach to estimate YoY inflation, period (12-month) average
inflation and their standard errors. In section 3 we present the results
of the application of the stochastic and the extended stochastic
approach for estimating MoM inflation, YoY inflation, and period
(12-month) average inflation along with their standard errors. In
section 4 we present the estimated average systematic change in relative
prices of different groups in the CPI basket of Pakistan.
'Concluding remarks follow in the last Section.
2. UNFOLDING THE STOCHASTIC APPROACH TO INDEX NUMBERS
The different ways to apply the stochastic approach to index
numbers give various forms of index numbers like Divisia, Laspeyres etc.
Since the Federal Bureau of Statistics (Pakistan's official
statistical agency) uses Laspeyres index formula for measuring inflation
in period t over the base period, we would like to confine the following
analysis to derive the Laspeyres index.
2.1. Derivation of Laspeyres Index Number (5)
Following conventional notations let p represents price and q
represent the quantity. We subscript these notations by it where i(i =
1,2, ..., n) represents commodity and t(t = 1,2, ... ... T) the time
(which is month, in this study). Under stochastic approach any observed
price change is a reading on the 'underlying' rate of
inflation and a random component ([[epsilon].sub.it]). If
[[gamma].sub.t] is the price index, relating expenditures in period t to
expenditures in the base period, then following the stochastic approach
we can write
[p.sub.it][q.sub.i0]=[[gamma].sub.t][q.sub.i0] + [[epsilon].sub.it]
t(t = 1,2, ... ... ..., T). (1)
We assume
E([[epsilon].sub.it]) = 0, Cov([[epsilon].sub.it],
[[epsilon].sub.jt]) =
[[sigma].sub.t.sup.2][p.sub.i0][q.sub.i0][[delta].sub.ij]
([[delta].sub.ij] is the Kronecker delta) (2)
In this way the index number theory has been related to regression
analysis as now we can estimate the rate of inflation in period t by
estimating the unknown parameter [[gamma].sub.t] in (1). Rearranging (1)
we get
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
To remove heteroscedasticity in the error term we transform
equation (1) into new form which gives homoscedastic variances in the
error term across all the n commodities in any particular time period t.
For this purpose we divide Equation (1) by [square root of [p.sub.i0]
[q.sub.i0]] and obtain
[y.sub.it] = [[gamma].sub.t][x.sub.i0] + [[eta].sub.it] (4)
Where [y.sub.it] = [P.sub.it][q.sub.i0]/[square root of [P.sub.i0]
[q.sub.i0]]; [x.sub.i0] = [square root of [p.sub.i0] [q.sub.i0] and
[[eta].sub.it] = [[epsilon].sub.it]/[square root of [p.sub.i0]
[q.sub.i0]]
Now assumptions in (2) after above transformation are
F([[eta].sub.it]) = 0 and Cov([[eta].sub.it], [[eta].sub.jt]) =
[[sigma].sub.t.sup.2] [[delta].sub.ij]
Now we can apply, say, the least squares to (4) to have an
estimator for inflation as below:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
We know [p.sub.i0][q.sub.i0]/[[summation].sup.n.sub.i=1][p.sub.i0][q.sub.i0] is the budget share of commodity i in the base period and if
we write it as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
Which is weighted average of the n price ratios (with base-period
budget shares being weights) and is the well-known Laspeyres price
index. With the help of this price index we can have inflation (MoM
and/or YoY) by using simple formulae as below:
Inflation (MoM) = ([[[??].sub.t]/[[??].sub.t-1]] - 1) x 100 (6)
Inflation (YoY) = ([[[??].sub.t]/[[??].sub.t-1]] - 1) x 100 (7)
Variance of the estimator in (5) is given by
Var([[??].sub.t]) =
[[sigma].sup.2.sub.t]/[[summation].sup.n.sub.i=1][x.sup.2.sub.10] (8)
The parameter [[sigma].sup.2.sub.t] can be estimated as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
By substitution the estimated parameter of [[sigma].sup.2.sub.t]
from (9) together with the values of [x.sub.i0] and [y.sub.it] in (8)
and rearranging we get
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
Thus, as the degree of relative price variability increases, the
variance of the estimated index also increases. This agrees with the
intuitive notion that when the individual prices move very
disproportionately, the overall price index is less well-defined [Logue
and Willet (1976)].
Now the question is can we have the estimated price index in (5)
and (10) and its estimated variance respectively. From (5) we can find
the estimated rate of inflation but here we cannot find the variance of
the estimated rate of inflation. For this purpose we have to proceed
with inflation from the start rather than the index.
2.2. Application of Stochastic Approach to Estimate Headline
Inflation and Its S.E.
Following the notations used above, if [[gamma].sub.t] is the price
index, relating expenditures in the current period to expenditures in
the base period, then, following the stochastic approach, we can write
[p.sub.it][q.sub.i0] = [[gamma].sub.t][p.sub.i0][q.sub.i0] +
[[epsilon].sub.it] t(t =1,2, ...., T) (11)
The base period can be somewhere in the distant past (say five year
back) and at any point in time we define headline (or, YoY) inflation as
percentage change in price index over the corresponding month last year
then
[[pi].sup.H.sub.t] = [[gamma].sub.t] -
[[gamma].sub.t-12]/[[gamma].sub.t-12]
From (11) we can get estimate of [[gamma].sub.t] only. For estimate
of [[gamma].sub.t-12] we write (11) as
[p.sub.it-12][q.sub.i0] = [[gamma].sub.t-12][p.sub.i0][q.sub.i0] +
[[epsilon].sub.it-12] t(t = 13,14,15 ...., T) (12)
Here again E([p.sub.it-12]/[p.sub.i0]) = [[gamma].sub.t-12], under
similar assumptions as in (2)
By subtracting (12) from (11) we have
[p.sub.it][q.sub.i0] - [p.sub.it-12][q.sub.i0] = ([[gamma].sub.t] -
[[gamma].sub.t-12][p.sub.i0][q.sub.i0] + [[epsilon].sub.it] -
[[epsilon].sub.it-12] (13)
Dividing (13) by E([p.sub.it-12]/[p.sub.i0]) and substituting
E([p.sub.it-12]/[p.sub.i0]) = [[gamma].sub.t-12] on right hand side, we
get
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
Where [e.sub.it] = [[epsilon].sub.it] -
[[epsilon].sub.it-12]/[[gamma].sub.t-12]. Again assuming that
E([e.sub.it]) = 0 and Cov{[e.sub.it], [e.sub.jt]) =
[[rho].sup.2.sub.t][p.sub.i0][q.sub.i0][[delta].sub.ij] (15)
and proceeding as in subsection 2.1 we divide (14) by [square root
of [p.sub.i0][q.sub.i0]] and get
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
From (5) and (12) we can write [[gamma].sub.t-12] = E
([p.sub.it-12]/[p.sub.i0]) = [[summation].sup.n.sub.i=1]
[w.sub.i0][[p.sub.it-12]/[p.sub.i0]], Thus (16) becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
If we take [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Under assumptions that E([[phi].sub.it])=0 and Cov([[phi].sub.it],
[[phi].sub.jt]) = [[??].sup.2.sub.t][[delta].sub.ij], for equation
[Y.sub.it] = [[pi].sup.H.sub.t][X.sub.i0] + [[phi].sub.it] (17)
Least square estimator of [[pi].sup.H.sub.t] is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
We knew this result from (7). The only benefit of the above process
is that now we can have an estimate of the standard error of headline
inflation as below:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)
The parameter [[??].sup.2.sub.t] can be estimated as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
By substitution of the estimated parameter [[??].sup.2.sub.t] in
(19) and rearranging we get
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)
Equation (20) shows that the variance of [[??].sup.H.sub.t]
increases with the degree of relative inflation variability. (6) Now we
move towards estimating the period average inflation and its standard
error.
2.3. Application of Stochastic Approach to Estimate Period Average
Inflation and Its S.E.
We know that period average (say 12 month average) inflation can be
calculated either by averaging the last 12 YoY inflation numbers or by
taking YoY inflation of the last 12-month (moving) averaged index
number. We would like to use the above result in subsection 2.1 for
estimating the 12-month average inflation, and those in subsection 2.2
for the standard error of period average inflation.
We have price index series as pit. If the 12-month averaged price
index series is denoted by [p.sup.A.sub.it] then following the results
in subsection 2.1, the estimate of YoY inflation of [p.sup.A.sub.it]
series will be
[[??].sup.A.sub.t] = [[summation].sup.n.sub.i=1]
[w.sub.i0][p.sup.A.sub.it]/[p.sub.i0] (21)
And thus
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)
Now for estimating the variance of the average inflation we use the
result in subsection 2.2 where we derived the standard error of YoY
inflation. If we replace the index with the average index in (20) we
will get the standard error of average YoY inflation, that is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)
3. MEASURING STANDARD ERRORS OF INFLATION IN PAKISTAN
In this section we present an application of the results described
and derived in the previous section by using the monthly data of prices
of 374 commodities covering the period July 2001-June 2010 for Pakistan.
(7) We present the estimated MoM inflation, YoY inflation, along with
their standard errors for Pakistan. As discussed above, there are
different ways to apply the stochastic approach to index numbers and
each culminates in different form of index numbers like Divisia,
Laspeyres etc. Just to compare our estimated results of inflation with
those from the Federal Bureau of Statistics we have used such
application of the stochastic approach which produces Laspeyres index
formula for measuring inflation in the current period over the base
period. Since the State Bank of Pakistan targets (12-month) average
inflation, particular attention has been paid to estimate period
(12-month) average inflation rate and its standard error, which is the
first empirical application of its type.
Table A1 in the Appendix presents the official rate of (monthly,
YoY and 12-month average) inflation and the estimated rate of (monthly,
YoY and 12-month average) inflation based on the stochastic approach
along with standard error of the estimate of inflation for Pakistan
economy based on the data for July 2001 to June 2010.8
Figures 1(a) to 1(c) present a scatter plot of inflation versus the
corresponding standard error for MoM, YoY and 12-month average
inflation; the solid line is the linear trend line.
[FIGURE 1(a) OMITTED]
[FIGURE 1(b) OMITTED]
[FIGURE 1(c) OMITTED]
From Figures 1(a) to 1(c) we can see that the standard error
increases with increasing inflation as depicted by the positive slope of
the trend line. This can be interpreted to mean that when inflation is
higher it becomes difficult to predict it. This agrees with the
intuitive notion that when the individual prices move very
disproportionately, the overall rate of inflation is less well defined.
These observations are in line with the past literature on the rate and
variability of inflation. (9) Figures 1(b) and 1(c) also answer the
question, "Why some central banks pursue (target) 12-month moving
average (YoY) inflation rate rather than monthly headline (YoY) rate of
inflation?" Some central banks use the 12-month average as the core
inflation. The answer is simple: the average (YoY) inflation is less
volatile than the headline inflation. This is evident from the lower
S.E. of 12-month moving average (YoY) inflation as shown in the figure
1(c) and compared to S.E. of monthly headline (YoY) inflation presented
in 1(b).
[FIGURE 2(a) OMITTED]
[FIGURE 2(b) OMITTED]
[FIGURE 2(c) OMITTED]
Figure 2 (a) to Figure 2(c) present the graph of all the three
types of inflation along with the respective 95 percent confidence band.
From Figure 2 (a) it is clear that the time when there is a jump in
inflation, as in April 2005 and May 2008, there is an increase in the
width of the confidence bands. Similarly we can note that in other
figures where the inflation is high, the width of confidence band is
also increased.
4. EXTENDED STOCHASTIC APPROACH AND THE SYSTEMATIC CHANGE IN
RELATIVE PRICES
As we discussed in Section 2, the various applications of the
stochastic approach to index numbers yield different forms of index
numbers like the Laspeyres etc. However, as criticised by Keynes (1930),
following this approach it is assumed that when prices change they
change equiproportionately and thus relative prices remain the same.
Clements and Izan (1987) responded to Keynes' criticism by
considering common trend change in all prices underlying the rate of
inflation separate from the systematic change in relative prices.
Following Clements and Izan (1987), if we take pit as the price of
commodity i(i = 1,2, ..., n) at time t(t = 1,2, ....., T) then price log
change [Dp.sub.it] = [p.sub.it] - log [P.sub.it-1] can be considered as
[Dp.sub.it] = [[alpha].sub.t] + [[beta].sub.i] + [[xi].sub.it] =
1,2, ..., n; and t = 1,2,...., T ... ... 24)
Where [[alpha].sub.t] is the common trend change in all prices (the
underlying rate of inflation) and [[beta].sub.i] is the change in
relative prices of commodity i. Assuming the random component of change
in prices, [[xi].sub.it], to be independent over commodities and time,
and the variances [Var([[xi].sub.it])] inversely proportional to
corresponding arithmetic averages of budget shares, Clements and Izan
(1987) showed that the least squares estimates of at and Pi are subject
to budget constraint (10) as given below:
[[??].sub.t] = [[summation].sup.n.sub.i=1]
[[bar.w].sub.i][Dp.sub.it] (25)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)
With respective variances of these estimators as below:
Var([[??].sub.t]) = [[theta].sup.2.sub.t]/(n - 1) (27)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)
Where [[theta].sup.2.sub.t] is the sum (over commodities) of
squares of estimated random component of price changes, that is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)
While it is obvious that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE
IN ASCII]
Our contribution in this section of the study is the application of
the Clements and Izan (1987) extended stochastic approach to index
numbers to Pakistan's monthly data of prices of 374 commodities
covering the period July 2001-June 2010. The extended stochastic
approach to index number is closer to Divisia price index. As in the
above Section 3, this approach also gives us the (trend) inflation rate
and its standard errors which are presented in Table A1 of the Appendix.
We can see that the estimated standard errors of the inflation rate
based upon extended stochastic approach to inflation are lower than
those based upon the stochastic approach for the MoM and YoY inflation.
It needs not be true in the case of period average inflation (because of
averaging effects).
In addition to inflation and its standard error, the Clements and
Izan (1987) extended stochastic approach also gives us the systematic
change in relative prices of each commodity in the basket. We have
applied this approach to prices of 374 commodities in the CPI basket of
Pakistan for the period of FY01 to FY 2010 to investigate the systematic
relative price changes. It may be difficult to extract any meaningful
result from the detailed presentation of systematic (MoM, and YoY, and
12-month average) change in relative prices of each of the 374
commodities. (11) However, it will be useful if we present the
systematic (MoM, and YoY, and 12-month average) change in relative
prices for various groups in the CPI basket as in Table A2 of the
Appendix. There are ten groups in the CPI basket of Pakistan as shown in
Table A2. It is clear from the table that coefficients of relative
prices of all groups are significantly different from zero. For
comparison we have also given the observed relative price changes as
measured from FBS price data for all the three cases: MoM, YoY and
12-month moving average. (12)
The estimated relative prices of'Food Beverages &
Tobacco' group are increased by highest percentage point for MoM
changes (0.18 percent). In case of YoY changes, we find 'Food
Beverages and Tobacco (FBT)' and 'Fuel and Lighting'
groups to exhibit increase in relative prices by 1.72 percent and 0.37
percent respectively. In the case of 12-month moving averages, we find
'FBT' and 'House Rent' groups to depict increase in
relative prices by 1.72 percent and 0.04 percent respectively. In all
the three cases, since the 'FBT' group turned out to have the
highest change in relative prices, we can say that inflation during most
of the FY01 to FY10 was FBT price change driven.
Interestingly, for each of the three cases of MoM, YoY and 12-month
moving average, the change in relative prices is found to be highest
(positive) for 'FBT' group and lowest (negative) for
'Recreation and Entertainment (RE)' group during FY01 to FY10.
The supply side factor(s) and/or elasticities of demand may be behind
this observed phenomenon as commodities in the FBT group are more prone
to supply shocks and tend to be less price elastic compared to those in
RE group.
Table A1 in the Appendix presents the official rate of (monthly,
YoY and 12-month average) inflation and the estimated rate of (monthly,
YoY and 12-month average) inflation based on stochastic as well as
extended stochastic approaches along with standard error of the
estimates of inflation for Pakistan based on the data for July 2001 to
June 2010. Numerically, the official and estimated inflation rates seem
different. But when we apply t-test we could not find official inflation
rate to be statistically different from any of the estimated inflation
rates based on stochastic as well as extended stochastic approaches.
(13) Which approach for measuring inflation is better? Obviously the
stochastic approach has an advantage as it estimates the standard errors
along with the inflation rate and is therefore preferable. Furthermore,
in the case of using extended stochastic approach we also get estimates
of systematic change in relative prices and their standard errors. The
confidence interval can be built around the estimated rate of inflation
for different useful purposes like wage bargaining.
5. CONCLUSION
In this study we estimate the standard errors of month on month and
year on year inflation rate using the stochastic approach of Selvanathan
and Selvanathan (2006) and the extended stochastic approach of Clements
and Izan (1987) based on individual prices of 374 commodities in CPI
basket of Pakistan for the period July 2001 to June 2010. We also
contribute to the literature by employing the stochastic approach to
index numbers by developing a mechanism to estimate the inflation rate
and its standard error for period average CPI. Based on this mechanism,
we estimate the standard error of 12-month moving average YoY inflation
rate for Pakistan for the period July 2003 to June 2010. We find that
the standard error of inflation increases with inflation rate in
Pakistan. Notwithstanding the fact that the 'higher the standard
error the higher the inflation rate', the estimated standard errors
of inflation rate based on extended stochastic approach are lower than
those based on the stochastic approach. Furthermore, for each of the
three cases of MoM, YoY and 12-month average, the change in relative
prices is found to be highest for 'Food Beverages and Tobacco'
group and lowest for 'Recreation and Entertainment' group
during FY01 to FY10.
APPENDIX
Table A1
Official Inflation Rate, Stochastic (Extended Stochastic)
Approach Related Estimates of Inflation and Respective
Standard Errors
Month on Month
Official Stochastic Extended
Inflation Approach Stochastic
Rate Estimates Approach
Month of Estimates of
inflation S.E. Inflation S.E.
Jul-03 0.57 1.01 0.35 0.89 0.24
Aug-03 0.66 0.76 0.36 0.62 0.15
Sep-03 0.60 0.53 0.35 0.32 0.18
Oct-03 1.47 1.57 0.57 1.20 0.33
Nov-03 0.60 0.69 0.34 0.74 0.14
Dec-03 0.90 1.17 0.47 1.12 0.17
Jan-04 -0 09 0.11 0.41 -0.07 0.27
Feb-04 -0.34 -0.48 0.38 -0.23 0.31
Mar-04 1.02 0.81 0.44 0.74 0.26
Apr-04 0.96 0.81 0.62 0.42 0.35
May-04 0.69 0.78 0.65 1.02 0.43
Jun-04 1.12 1.00 0.29 1.12 0.20
Jul-04 1.38 1.58 0.37 1.20 0.25
Aug-04 0.59 0.64 0.39 0.62 0.14
Sep-04 0.37 0.27 0.30 0.28 0.12
Oct-04 1.19 1.19 0.51 0.76 0.14
Nov-04 1.12 0.99 0.50 1.05 0.21
Dec-04 -0.85 -0.97 0.70 -0.05 0.38
Jan-05 0.97 1.04 0.23 0.95 0.19
Feb-05 0 99 1.03 0.22 0.83 0.17
Mar-05 1.29 1.56 0.55 0.84 0.27
Apr-05 1.74 2.10 1.01 0.94 0 41
May-05 -0.44 -0.84 0.88 0.16 0.31
Jun-05 0.10 -0.28 0.42 0.74 0.19
Jul-05 1.62 1.55 0.39 1.06 0.18
Aug-05 0.04 0.12 0.44 0.12 0.23
Sep-05 0.50 0.44 0.35 0.34 0.15
Oct-05 0.94 0.53 0.43 0.51 0.16
Nov-05 0.76 0.94 0.53 0.93 0.18
Dec-05 -0.27 -0.33 0.44 -0.10 0.23
Jan-06 1.20 1.28 0.37 0.96 0.31
Feb-06 0.33 0.42 0.45 0.69 0.32
Mar-06 0.23 0.25 0.25 0.16 0.17
Apr-06 1.02 1.09 0.67 0.68 0.29
May-06 0.45 0.32 0.74 0.34 0.22
Jun-06 0.59 0.49 0.37 0.94 0.30
Jul-06 1.61 1.64 0.40 1.09 0.24
Aug-06 1.25 1.38 0.44 0.97 0.15
Sep-06 0.32 0.18 0.32 0.38 0.11
Oct-06 0.36 0.44 0.53 0.20 0.22
Nov-06 0.73 1.10 0.75 0.68 0.28
Dec-06 0.47 0.88 0.67 0.50 0.33
Jan-07 -0.88 -1 10 0.81 -0.36 0.52
Feb-07 1.04 1.19 0.36 0.95 0.21
Mar-07 0.49 0.27 0.51 0 18 0.31
Apr-07 0.31 0.07 0.80 -0.05 0.31
May-07 0.92 0.97 0.97 0.85 0.33
Jun-07 0.20 -0.09 0.46 0.92 0.20
Jul-07 1.01 1.08 0.39 1.10 0.26
Aug-07 1.32 1.46 0.45 1.06 0.23
Sep-07 2.13 2.32 0.57 1.54 0.23
Oct-07 1.23 1.27 0.58 0.90 0.17
Nov-07 0.14 -0.15 0.59 0.40 0.26
Dec-07 0.58 0.74 0.49 0.85 0.23
Jan-08 1.91 1.94 0.35 1.58 0.25
Feb-08 0.49 0.18 0.55 0.13 0.41
Mar-08 3.08 3.22 0.88 2.35 0.34
Apr-08 3.04 3.17 1.18 2.60 0.43
May-08 2.69 2.79 0.83 2.73 0.41
Jun-08 2 10 1.91 0.35 2.17 0.24
Jul-08 3.34 3.33 0.44 3 03 0.39
Aug-08 2.14 2.33 0.53 1.84 0.29
Sep-08 0.97 0.86 0.41 0.79 0.20
Oct-08 2.12 2.15 0.57 1.72 0.26
Nov-08 -0.12 -0.29 0.53 0.62 0.40
Dec-08 -0.50 -0.25 0.48 0.23 0.17
Jan-09 -0.42 -0.55 0.39 -0.13 0.31
Feb-09 0.95 1.22 0.34 1.15 0.23
Mar-09 1.37 1.94 0.70 0.95 0.25
Apr-09 1.41 1.10 0.85 0.98 0.29
May-09 0.23 0.04 0.82 0.40 0.40
Jun-09 0.99 0.70 0.36 1.32 0.27
Jul-09 1.54 1.61 0.38 1.30 0.22
Aug-09 1.70 1 66 0.51 1.30 0.27
Sep-09 0.45 0.38 0.38 0.39 0.19
Oct-09 0.95 0.89 0.45 0.66 0.15
Nov-09 1.39 1.27 0.48 1.35 0.22
Dec-09 -0.49 -0.36 0.51 0.04 0.23
Jan-10 2.42 2.47 0.31 2.28 0.29
Feb-10 0.39 0.37 0.28 0 58 0.22
Mar-10 1.25 1.29 0.22 0.99 0.16
Apr-10 1.73 2.07 0.71 1 85 0.42
May-10 0.06 -0.19 0.76 0.17 0.20
Jun-10 0.65 0.34 0.37 0.78 0.17
Headline (Year on Year)
Official Stochastic Extended
Rate of Approach Stochastic
Inflation Estimates Approach
Month of Estimates of
Inflation S.E. Inflation S.E.
Jul-03 1.41 1.72 0.60 1.80 0.54
Aug-03 1.76 2.03 0.63 2.04 0.52
Sep-03 2.18 2.34 0.75 2.21 0.54
Oct-03 3.51 4.94 0.56 3.47 0.50
Nov-03 4.22 5.08 0.64 4.29 0.57
Dec-03 5.41 6.92 0.83 5.43 0.56
Jan-04 5.15 6.67 0.98 5.26 0.58
Feb-04 4.31 5.43 0.82 4.62 0.59
Mar-04 5.34 6.19 0.63 5.52 0.51
Apr-04 5.99 6.39 0.59 6.02 0.48
May-04 7.04 7.57 0.60 7.03 0.50
Jun-04 8.45 9 10 0.67 7.89 0.53
Jul-04 9.33 9.72 0.78 8.21 0.56
Aug-04 9.25 9.59 0.75 8.20 0.52
Sep-04 9.00 9.30 0.76 8.16 0.55
Oct-04 8.70 8.89 0.69 7.72 0.49
Nov-04 9.26 9.22 0.66 8.04 0.51
Dec-04 7.37 6.91 0.73 6.87 0.54
Jan-05 8.51 7.89 0.90 7.88 0.63
Feb-05 9.95 9.53 0.80 8.95 0.66
Mar-05 10.25 10.35 0.76 9.05 0.60
Apr-05 11.10 11.76 1.15 9.56 0.61
May-05 9.84 9.96 0.71 8.70 0.51
Jun-05 8.74 8.57 0.51 8.32 0.46
Jul-05 8 99 8 54 0.53 8.17 0.48
Aug-05 8.40 7.98 0.52 7.67 0.46
Sep-05 8.53 8.17 0.53 7.73 0.48
Oct-05 8.27 7 46 0.75 7.48 0.55
Nov-05 7.89 7.41 0.70 7.36 0.54
Dec-05 8.51 8.10 0.54 7.30 0.45
Jan-06 8.76 8.37 0.54 7.32 0.40
Feb-06 8.05 7.71 0.63 7.17 0.48
Mar-06 6.91 6.32 0.64 6.49 0.51
Apr-06 6 16 5.27 0.96 6.24 0.58
May-06 7.12 6.51 0.76 6.42 0.58
Jun-06 7.65 7.33 0.51 6 62 0.43
Jul-06 7.63 7.42 0.61 6.65 0.52
Aug-06 8.93 8.77 0.62 7 50 0.48
Sep-06 8.73 8.49 0.61 7.55 0.51
Oct-06 8.11 8.39 0.63 7.24 0.46
Nov-06 8.07 8.55 1.09 6.98 0.52
Dec-06 8.88 9.88 1.60 7.59 0.73
Jan-07 6.64 7.29 1.04 6.26 0.57
Feb-07 7.39 8 12 1.23 6.53 0.64
Mar-07 7.67 8.14 1.02 6.55 0.66
Apr-07 6.92 7.05 0.85 5.82 0.58
May-07 7.41 7.74 0.98 6.32 0.72
Jun-07 7.00 7.12 0.67 6.30 0.61
Jul-07 6.37 6.53 0.67 6.31 0.62
Aug-07 6.45 6.61 0.60 6.40 0 55
Sep-07 8.37 8.89 0.79 7.57 0.57
Oct-07 9.31 9.79 0.85 8.27 0.61
Nov-07 8.67 8.44 0.93 8.00 0.67
Dec-07 8.79 8.29 1.29 8.35 0.77
Jan-08 11.86 11.62 1.20 10.29 0.82
Feb-08 11.25 10.50 1.36 9 46 0.88
Mar-08 14.12 13.75 1.54 11.64 0.86
Apr-08 17.21 17.28 1.05 14.29 0.73
May-08 19.27 19.40 1.36 16.17 0.87
Jun-08 21.53 21.79 1.38 17.43 0.92
Jul-08 24.33 24.49 1.33 19.36 0.79
Aug-08 25.33 25.57 1.32 20.13 0.76
Sep-08 23.91 23.78 1.19 19.38 0.71
Oct-08 25.00 24.85 1.35 20.19 0.79
Nov-08 24.68 24.68 1.10 20.41 0.66
Dec-08 23.34 23.45 0.98 19.79 0.59
Jan-09 20.52 20.43 0.75 18.08 0.55
Feb-09 21.07 21.69 1.13 19.10 0.70
Mar-09 19.07 20.17 1 83 17.69 0.87
Apr-09 17.19 17.76 1.10 16.07 0.75
May-09 14.39 14.61 0.88 13.74 0.69
Jun-09 13.14 13.25 0.84 12.89 0.73
Jul-09 11.17 11.37 0.85 11.17 0.72
Aug-09 10.69 1064 0.82 10.63 0.71
Sep-09 10.12 10.11 0.78 10.23 0.68
Oct-09 8.87 8.74 0.66 9.17 0.63
Nov-09 10.51 1045 0.69 9.90 0.55
Dec-09 10.52 10.32 0.72 9 72 0 56
Jan-10 13.68 13.68 0.81 12.13 0.58
Feb-10 13.04 12.73 0.77 11.56 0.58
Mar-10 12.91 12.01 1.09 11.61 0.66
Apr-10 13.26 13.09 0.81 12.48 065
May-10 13.07 12.83 0.79 12.24 0.64
Jun-10 12.69 12.43 0.74 11.70 0.62
12-month moving average
Official Stochastic Extended
Rate of Approach Stochastic
Inflation Estimate Approach
Month of Estimates of
Inflation S.E. Inflation S.E.
Jul-03 2.89 3.03 0.16 1.54 0.48
Aug-03 2.73 2.68 0.14 1.52 0.48
Sep-03 2.60 2.50 0.13 1.50 0.50
Oct-03 2.60 2.37 0.12 1.64 0.50
Nov-03 2.70 2.52 0.13 1.89 0.49
Dec-03 2.87 2.68 0.14 2.22 0.46
Jan-04 3.02 3 06 0.16 2.54 0,42
Feb-04 3.09 3.29 0.17 2.79 0.40
Mar-04 3.35 3.73 0.19 3.21 0.38
Apr-04 3.66 4.07 0.21 3 56 0.37
May-04 4.03 4.56 0.23 4.08 0.36
Jun-04 4.57 5.26 0.27 4.70 0.36
Jul-04 5.23 5.94 0.30 5.23 0.36
Aug-04 5.86 6.58 0.34 5.75 0.37
Sep-04 6.43 7.03 0.36 6.27 0.40
Oct-04 6.86 7.41 0.38 6.62 0.41
Nov-04 7.29 7.77 0.40 6.94 0.42
Dec-04 7.45 7.79 0.40 7.03 0.40
Jan-05 7.72 7.90 0.40 7.21 0.38
Feb-05 8.19 8.23 0.42 7.54 0.39
Mar-05 8.60 8.56 0.44 7.81 0.40
Apr-05 9.03 9.00 0.46 8.11 0.40
May-05 9.26 9.18 0.47 8.22 0.39
Jun-05 9 28 9.13 0.47 8.27 0.39
Jul-05 9.25 9.04 0.46 8.28 0.38
Aug-05 9.17 8.90 0.46 8.24 0.37
Sep-05 9.13 9.12 0.47 8 21 0.36
Oct-05 9.09 9.01 0.46 8.18 0.37
Nov-05 8.97 8.84 0.45 8.08 0.38
Dec-05 9.06 8.93 0.46 8.11 0.38
Jan-06 9.08 8.96 0.46 8.06 0.39
Feb-06 8.92 8.81 0.45 7.93 0.40
Mar-06 8.64 8.50 0.44 7.73 0.41
Apr-06 8.23 7.98 0.41 7.45 0.42
May-06 8.01 7.71 0.40 7.28 0.43
Jun-06 7.92 7.61 0.39 7.14 0.43
Jul-06 7.81 7.52 0.39 7.01 0.42
Aug-06 7.86 7.59 0.39 7 00 0 40
Sep-06 7.88 7.45 0.38 6.97 0.39
Oct-06 7.87 7.49 0.38 6.96 0.38
Nov-06 7.89 7.61 0.39 6.97 0.37
Dec-06 7.92 7.76 0.40 7.03 0.38
Jan-07 7.74 7.68 0.39 6.96 0.37
Feb-07 7.69 7.71 0.40 6.91 0.35
Mar-07 7.75 7.85 0.40 6.93 0.34
Apr-07 7.81 8.01 0.41 6.91 0.33
May-07 7.83 8.10 0.42 6.89 0.35
Jun-07 7.77 8.07 0.41 6.87 0.36
Jul-07 7.66 7.99 0.41 6.84 0.36
Aug-07 7.45 7.80 0.40 6.74 0 38
Sep-07 7.43 7.84 0.40 6.75 0.41
Oct-07 7.54 7.99 0.41 6.85 0.44
Nov-07 7.60 7.97 0 41 6.90 0.45
Dec-07 7.60 7.84 0.40 6.92 0.47
Jan-08 8.04 8.21 0.42 7.27 0.51
Feb-08 8.36 8.42 0.43 7.50 0.54
Mar-08 8.91 8 89 0.46 7.95 0.56
Apr-08 9.78 9.74 0.50 8.67 0.57
May-08 10.78 10.74 0 55 9.55 0.58
Jun-08 12.00 11.97 0.61 10.53 0.59
Jul-08 13.51 13.48 0.69 11.67 0.60
Aug-08 15.10 15.09 0.77 12.86 0.61
Sep-08 16.42 16.34 0.84 13.86 0.62
Oct-08 17.75 17.62 0.90 14.84 0.64
Nov-08 19.09 18.99 0 97 15.89 0.63
Dec-08 20.29 20.24 1.04 16.84 0.60
Jan-09 20.97 20.94 1.07 17.44 0.56
Feb-09 21.75 21.84 1.12 18.19 0.55
Mar-09 22.11 22.34 1.15 18.59 0.54
Apr-09 22.04 22.32 1.14 18.68 0.52
May-09 21.55 21.81 1.12 18.39 0.48
Jun-09 20.77 21.01 1.08 17.90 0.45
Jul-09 19.60 19.84 1.02 17.14 0.44
Aug-09 18.33 18.38 0.94 16.29 0.45
Sep-09 17.15 17.05 0.87 15.49 0.48
Oct-09 15.79 15.61 0.80 14.56 0.51
Nov-09 14.65 14.35 0.74 13.69 0.51
Dec-09 13.65 13.19 0.68 12.87 0.51
Jan-10 13.15 12.59 0.65 12.42 0.52
Feb-10 12.57 11.84 0.61 11.87 0.52
Mar-10 12.12 11.15 0.57 11.42 0.51
Apr-10 11.84 10.76 0.55 11.16 0.51
May-10 11.75 10.57 0.54 11.06 0.51
Jun-10 11.73 10.43 0.54 11.01 0.51
Source: Authors' calculations, except the official inflation rate for
which the source is Pakistan Bureau of Statistics.
Table A2
Group-wise Change in Relative Price (July 01-June 10)
Month on Month Inflation
Observed Estimated S.E.
Weight Change in Change in
in CPI Relative Relative
Group Basket Price (%) Price (%)
Food Beverages
& Tobacco 0.403 0.14 0.18 0.001
Apparel, Textile
& Footwear 0.061 -0.26 -0.21 0.004
House Rent 0.234 -0.06 -0.01 0.001
Fuel & Lighting 0.073 0.03 -0.18 0.003
Household
Furniture &
Equipment 0.033 -0.24 -0.18 0.008
Transport &
Communication 0.073 -0.06 -0.14 0.003
Recreation &
Entertainment 0.008 -0.45 -0.51 0.035
Education 0.035 -0.16 -0.20 0.008
Cleaning,
Laundry &
Personal
Appearance 0.059 -0.17 -0.17 0.005
Medicare 0.021 -0.28 -0.27 0.140
Headline (Year on Year)
Observed Estimated S.E.
Change in Change in
Relative Relative
Group Price (%) Price (%)
Food Beverages
& Tobacco 1.62 1.72 0.01
Apparel, Textile
& Footwear -3.38 -3.15 0.09
House Rent -0.51 -0.36 0.02
Fuel & Lighting 0.73 0.37 0.08
Household
Furniture &
Equipment -2.89 -2.37 0.18
Transport &
Communication -0.34 -1.06 0.08
Recreation &
Entertainment -6.20 -6.33 0.75
Education -1.77 -1.48 0.75
Cleaning,
Laundry &
Personal
Appearance -2.08 -2.21 0.01
Medicare -3.77 -2.86 0.29
12-month moving average
Observed Estimated S.E.
Change in Change in
Relative Relative
Group Price (%) Price (%)
Food Beverages
& Tobacco 1.47 1.72 0.01
Apparel, Textile
& Footwear -3.18 -3.22 0.07
House Rent -0.55 0.04 0.02
Fuel & Lighting 0.99 -0.44 0.06
Household
Furniture &
Equipment -2.59 -2.16 0.14
Transport &
Communication -0.09 -1.37 0.06
Recreation &
Entertainment -6.17 -6.78 0.55
Education -1.87 -1.58 0.13
Cleaning,
Laundry &
Personal
Appearance -2.04 -2.14 0.07
Medicare -3.34 -3.42 0.22
Source: Authors' calculations.
REFERENCES
Balk, B. M. (1980) A Method for Constructing Price Indices for
Seasonal Commodities. Journal of the Royal Statistical Society, Series A
142, 68-75.
Bryan, M. F. and S. G. Cecchetti (1993) The Consumer Index as a
Measure of Inflation. Federal Reserve Bank of Cleveland Economic Review
4, 15-24.
Clements, K. W. and H. Y. Izan (1981) A Note on Estimating Divisia
Index Numbers. International Economic Review 22, 745-747.
Clements, K. W. and H. Y. Izan (1987) The Measurement of Inflation:
A Stochastic Approach. Journal of Business and Economic Statistics 5:3,
339-350.
Clements, K. W. and E. A. Selvanathan (2007) More on Stochastic
Index Numbers. Applied Economics 39, 605-611.
Clements, K. W., H. Y. Izan, and E. A. Selvanathan (2006)
Stochastic Index Numbers: A Review. International Statistics Review
74:2, 235-270.
Crompton, P. (2000) Extending the Stochastic Approach to Index
Numbers. Applied Economics Letters 1, 367-371.
Diewert, W. E. (1995) On the Stochastic Approach to Index Numbers.
Department of Economics, University of British Columbia. (Discussion
Paper No. 31).
Edgeworth, F. Y. (1888) Some New Methods of Measuring Variation in
General Prices. Journal of the Royal Statistical Society 51:2, 346-368.
Frisch, R. (1936) The Problem of Index Numbers. Econometrica 4:1,
01-38.
Jevons, W. S. (1865) On the Variation of Prices and the Value of
the Currency since 1782. Journal of Statistical Society of London 28:2,
294-320.
Keynes, J. M. (1930)/! Treatise on Money. Vol. 1. London:
Macmillan.
Logue, D. E. and T. D. Willet (1976) A Note on the Relation between
the Rate and Variability of Inflation. Economica 43:170, 151-158.
Selvanathan, E. A. (1989) A Note on the Stochastic Approach to
Index Numbers. Journal of Business and Economic Statistics 7:4, 471-474.
Selvanathan, E. A. (2003) Extending the Stochastic Approach to
Index Numbers: A Comment. Applied Economics Letters 10, 213-215.
Selvanathan, E. A. and D. S. P. Rao (1994) Index Numbers: A
Stochastic Approach. London: Macmillan.
Selvanathan, E. A. and S. Selvanathan (2004) Modelling the
Commodity Prices in the OECD Countries: A Stochastic Approach. Economic
Modelling 21:2, 233-247.
Selvanathan, E. A. and S. Selvanathan (2006) Measurement of
Inflation: An Alternative Approach. Journal of Applied Economics 2,
403-418.
Selvanathan, E. A. and S. Selvanathan (2006b) Recent Developments
in the Stochastic Approach to Index Numbers. Applied Economics 38:12,
1353-1362.
(1) In rest of the paper we would prefer to use the term
'inflation' instead of the 'rate of inflation' or
'inflation rate,' for brevity.
(2) This 'underlying' rate of inflation need not be
confused with (a completely different) concept of 'core
inflation' being used by some central banks to see long-term trend
in change in general price level (sans temporary, short term,
non-monetary, and supply side related changes in inflation). Core
inflation is altogether a different measure of change in prices of a
shrunk set of commodities which are more linked with demand side
compared to supply side (exclusion based measures like non-food non
energy inflation rate) or trimmed set of commodities prices of which are
not highly volatile (like 20 percent trimming--those commodities which
show extreme price changes--based measures of inflation).
(3) Clements, Izan and Selvanathan (2006) presented a review on the
stochastic approach to index number theory.
(4) As mentioned earlier, we have not estimated the standard error
of the core inflation rate measures like those exclusion-based or
trimming-based. However, our working on the 12 month average inflation
rate may be viewed as an exercise on the core inflation (because 12
month average inflation rate can be used as core inflation measure
because it smoothes out fluctuations).
(5) This sub-section (2.1) is mostly based upon Selvanathan and
Selvanathan (2006b).
(6) Above procedure can be used to estimate the MoM inflation and
its standard error.
(7) Prices, for construction of consumer price index (CPI), are
collected by Pakistan Bureau of Statistics (the central statistical
agency of the Government of Pakistan) on monthly basis. In August 2011,
while changing the base year for CPI from FY 2001 to FY 2008, PBS also
expanded the coverage of goods/services in the CPI basket by increasing
the number of commodities from 374 to 487. In this study, we have used
the previous base (FY 2001) dataset.
(8) First 12 observations are lost in the YoY inflation calculation
and next 12 are consumed in calculating the 12 month average. Thus, the
results in the Table I start from July 2003 instead of July 2001.
(9) The inflation may become less predictable at higher inflation
rate if government aims stabilising prices rather than stabilising
expectations [Logue and Willet (1976)].
(10) Budget share weighted average of the systematic component of
relative price change is zero.
(11) Detailed results can be obtained from the authors, if desired.
(12) We can see from the Table A2 in the Appendix that the weighted
average of the relative prices is zero in each of the observed and
estimated case, which should be.
(13) The results of t-test are not reported in the paper to save
the space. However, those can be obtained from the authors, if required.
Javed Iqbal <
[email protected]> is Research Economist
at the State Bank of Pakistan (SBP) Karachi. M. Nadim Hanif
<
[email protected]> is Additional Director in the
State Bank of Pakistan (SBP) Karachi.
Authors' Note: This paper is part of PhD dissertation of the
first author who is thankful to Asad Zaman for supervision of this work.
Views expressed in this paper are those of the authors may not be
attributed to SBP where they are employed. Authors are thankful to
Riazuddin, Ali Choudhary, Mahmood ul Hassan Khan, Muhammad Rehnian,
Jahanzeb Malik and two anonymous referees for their comments on an
earlier draft of this paper.
