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Roomer Hypothesis of Sacrifice Ratio: Asymmetric Analysis in Case Of Pakistan.


Roomer (1993) portrayed a hypothesis of sacrifice ratio; in order alter the trend of the inflation the economy has to bear some cost of slowing down in economic growth. Inflation is a continuous headache for the policy makers along to keep stable economic growth. Mainly there are two main subject of occurring inflation in an economy, the first one is demand pull inflation and the second one is the cost push inflation. Throughout the world, from developed to under developed countries it has been equally revealed that the sacrifice ratio being remain different to different country Naqvi and Khan (1989), Barro (1995), Lim and Pap (1997), Kuijs (1998), Liu.and Adedeji (2000), Abdullah and Khalim (2009) and.Khan and.Gill (2010). The two school of thoughts, highlighted this issue, the first one is a structuralist who emphasized that inflation is part and parcel for advancing the economic growth whereas the monetarist it only could determine the economic growth kydland and Prescot (1977) and Barro and Gordon (1983). A common thought has been developed that in a low sacrifice ratio; by lowering a less of economic growth the inflation is checked or maintained at equilibrium level , is more preferred over the high sacrifice ratio in an economy Malik and Chowdhry (2001) and Mubarik (2005). In order to check this argument the econometrician proposed threshold level, which discriminate the sacrifice ratio as before threshold level and after threshold level. This threshold level will show the sacrifice ratio in two fold, such as, before threshold level the cost of lowering the economic growth to check the inflation and after threshold point the cost of reducing the economic growth to alter the behavior of inflation in the economy.

As being key factor of the economy, inflation is discussed a lot by policy makers. The appropriate trend of inflation which fosters the economic growth is searched by researcher constantly. In last decade, a plenty of studies have been conducted in this context by incorporating the threshold level of inflation in the model along threshold cointegration.Munir (2009), .Burdekin (2000) and.Gillman & Kejak (2000).

In case of Pakistan, a single digit inflation has been claimed by Naqvi and Khan. (1989) to advance the economic growth positively. Malik and Chowdhury. (2001) analyzed the short and long run relationship of sacrifice ratio for the South Asian countries through cointegration model taking panel data test. They claimed there occur a positive association between economic growth and inflation for Pakistan.

Mubarik (2005) analyzed 9% threshold level in case of Pakistan by using data 1973 to 2005 by applying linear test of cointegration in this study. Hussain (2005) investigated and revealed belt of inflation comprised between 4-6 percent level using 1973 to 2005 time series data and claimed that before and after this belt there would be deterioration of economic growth a high sacrifice ratio, further many studies have been done on this issue resulting the low and high sacrifice ration between economic growth and inflation see Khan (2007) Abdullah and Kalim. (2009) and Bashir (2011).

As earlier, the Engle and Granger (1982) and Granger and Weiss (1983) presented the key concept of conintegration and afterward advocated by Johansen (1991) and Juselious (1990). The key assumption of their test reveals that the disequilibrium among independent and dependent variables, move towards long-run equilibrium almost in every period. On contrary to it, Balke and Fomby (1997) Enders and Siklos (2001) and Hansen and Soe (2002) argued that such fluctuations are not subject to return back to equilibrium in every time period and proposed concept of threshold cointegration. They claimed that in case of nonlinearity of the data the linear cointegration is weakened as not incorporating the nonlinearity in the model, rather threshold cointegration produce more strengthen results of long run association among variables. Mubarik (2005) and Hussain (2005) calculated the threshold level of inflation in case of Pakistan fonly but they used linear.cointegraiton and linear error correction model. In regards of Pakistan, in among all other studies, as to best of my findings, there is not a single study that incorporated threshold level neither the nonlinearity is determine among the sacrifice ratio as well as threshold cointegration. That create a clear gap in literature of Pakistan case which is done up through this study.

The study pursued to: Estimate the nonlinearity in inflation, economic growth and exchange rate. Estimate the Threshold cointegration among variables. Estimate the Sacrifice ratio as in more persistent regime and less persistent regime.


The estimation have been done by using quarterly time series data ranging between 1980 Q1 to 2016 Q4 for the variables inaltion rate ([[pi].sup.5.sub.t]) , economic growth ([[gamma].sup.5.sub.t]) and exchange rate ([er.sup.4.sub.t]). The source of data is international financial statistics (IFS) along various issue of economic survey of Pakistan.

In order to see the sacrifice ratio for more persistent and less persistent regime TAR model has been used Khanand..Senhadji (.2001), Ahmed and Mortaza (2005) and Munir (2009) constructed as:

[y.sub.t.] =.[a.sub.1] + [a.sub.2][[pi].sub.t]+[a.sub.3][er.sup.4.sub.t] + [1.sup.+.sub.1t] + [[a.sub.3] + [a.sub.4][[pi].sub.t]+[a.sub.5][er.sup.5.sub.t]]. + [1.sup.-.sub.1t] + [[epsilon].sub.1] (1)

where [1.sup.+.sub.1t]. = 1

if [n.sub.t] [less than or equal to] [gamma]. (threshold level),.

Else zero. Similarly [1.sup.+.sub.2t] = 1. if [y.sub.t] [less than or equal to] [TAU], Ele. zero,. and9 [[epsilon].sub.t1], 9[[epsilon].sub.t2]~iid.

In case of existence of nonlinearity in the series of a variable, Balke.and Fomby.(1997) stressed that this fluctuation reverting back towards equilibrium is not occur in every period rather it reaches a threshold level then it revert back. Hansen (1999) developed a test for nonlinearity in case of one threshold and two threshold level in a single series such as:

[Y.sub.t] = [a'.sub.1][X.sub.t-1][I.sub.1t](1[[gamma].sup.1.sub.2],[[phi].sup.1.sub.2]) + ... + [a'.sub.m][X.sup.7.sub.t-m][]([gamma],[[phi].sup.1.sub.2]) + [[epsilon].sub.t] (2)

where [gamma] = ([[gamma].sub.1], ...,[[gamma].sub.m-1]) with [[gamma].sub.1]<[[gamma].sub.2]<... < [[gamma].sub.m-1], and [I.sub.ij](y,[[phi].sup.1.sub.2] ) = I ([[gamma].sup.l.sub.j-i] < 4[Y.sup.3.sub.t-d2] [less than or equal to][[gamma].sup.9.sub.j]) .where an indicator function.

And [mathematical expression not reproducible]. Here, [e.sub.m] is n x 1 vector of residuals and computed through least square and the RSS's are represented by [S.sub.m] = [e'.sub.m][e.sub.m]. Finally, LS test which test hypothesis of AR (.1) vs TAR (.mi) is as:

[F.sub.jk] = n([S.sub.j]-[S.sub.k]/[S.sub.k]) (3)

Where Sj the RSS for SETAR (1) and Sk for TAR (m). It is attributed as Wald/ F-test test statistic for null hypothesis to AR.(1) model against TAR(mi) and after confirming the long-run association,.Hansen.and.Soe.(2002) presented nonlinear VECM, such as:

[mathematical expression not reproducible] (4)

Where [DELTA] is difference operator, [X.sub.t-1] presents k x 1 matrix, [gamma] is threshold point and ut supposed to be 2 x 1 and [summation] = E([u.sub.t],[u'.sub.t]) is covariance finite matrix. A constrain is put by Hansen and Seo (2002) over threshold value as [[pi].sub.0] < P([w.sub.t-i] [less than or equal to] [gamma]) [less than or equal to] 1 - [[pi].sub.0], where [[pi].sub.0] = 0.05 and 1 - [[pi].sub.0] = 0.95. Through the MLE approach the model (2) estimation is as:

[mathematical expression not reproducible] (5)

Whe [mathematical expression not reproducible]

Hansan and Seo (2002) by holding ([beta],[gamma]) fixed has concentrated out ([A.sub.1],[A.sub.2],[summation]), then estimate constrained MLE for parameters ([beta],[gamma]) . MLE ([beta],[gamma]) minimized the log|[summation] ([beta], [gamma]) | in (10) as subject to normalization of equation (9) and it can be re-written as:

[mathematical expression not reproducible] (11)

[mathematical expression not reproducible] (12)

Hansen and Seo (2002) inference based when cointegrated vectors are known for null hypothesis of linear cointegration against alternative of threshold cointegration ( as given in equation (7)) and no cointegration against threshold cointegration. The Sup-LM test is as:

Sup - Wald = [sup.sub.YL<[gamma][less than or equal to][gamma]u]Wald ([beta]0,[gamma]) ... ... ... ..(14)

Where [[beta].sub.0] is the known value of [beta] and obtained through running simple ordinary least square model for inflation and economic growth. The Sup-Wald test statistic is for null hypothesis of linear cointegratoin against threshold cointegration.

Finally, Hansen and Soe (2002) the TVEC model is defined as:

[mathematical expression not reproducible] (19)

Where, H/L is for high/low regimes. a and b are constants and coefficients matrix. The [w.sub.t-1] is the ECM and [epsilon]~iid.


The key property of time series data is stationarity, which is checked through ADF () unit root test as well as Philip-perron test. The estimated table is as:

In above Table 1, all the variables are stationary of same order, integrated of order one. Now, moving forward towards estimate, the estimated model for null of no threshold and one threshold estimate is as follows:

From above Table 2, it is shown that there is rejection of null hypothesis of no threshold level in the growth rate series and accept the null hypothesis of threshold level. Further, accept the null hypothesis of one threshold level rather two threshold estimate in the series. Through F-test statistics the null hypothesis are accepted and rejected having confidence interval of 95%. Now, by confirming the threshold estimate and nonlinearity, the estimation advancing towards the threshold cointegration as explained in the methodology. The estimated model is as:

In the above Table 3, the test statistic for the null hypothesis of linear cointegration rejected both through Sup LM statistics (25.702) and through the residual and fixed regressors bootstrap and accepted the alternative of the threshold cointegration. The estimated threshold parameter is 7.1 in inflation series, there is drift term included in the model along one lag length at 10% trim values to avoid potential threshold. Next, TVECM model is estimated to see how much speed of adjustment parameter takes variation in different regime and takes how much time to revert back towards the equilibrium of steady state level in the economy. The estimated model is as:

In above Table 4, TVECM model is depicted that shows, in regime one or in less persistent phase, the value of the speed of adjustment is -0.54 which is significant, negative and below one, thus fulfilling the properties of the convergent equilibrium in the long run. It shows the gap between growth rate as a dependent variable and inflation rate, exchange rate independent variables will converge to the equilibrium with the speed of 54% per year. On the other side, in more persistent regime, the value of the speed of adjustment variable is -0.74, that is also significant, negative and below one. Thus, it's also showing the convergent in the long run towards the equilibrium but with a little bit different speed that is 74% per year. From the above estimation it is shown that economic growth reacting differently, in different phases to the inflation and exchange rate in the economy. It has a strong and power of reverting equilibrium in the high persistent regime as compare to the less persistent regime. Finally, over main nonlinear regression model is estimated with the two regimes as discriminated by inflation as before 7.1% and after 7.1% keeping economic growth as a dependent, as:

In above Table 5, nonlinear regression model is showing different sacrifice ratios, such as it is shown that in less persistent regime the cost of slowing down of economic growth in order to check the inflation trend is different to high persistent regime. In less persistent regime or lower regime, there is a positive relationship or sacrifice ratio is showing the common trend. The coefficient of inflation rate is 0.313 statistically significant and having positive sign. It shows that in less persistent regime there is increase in economic growth due to increase in inflation rate. Whereas, in high regime, the sacrifice ratio is negative and is showing dissimilar interest between inflation and economic growth. It shows in more persistent regime the economic growth is associated negatively to the inflation rate in the economy and deteriorating the economic growth by 0.169% with a unit increase in the inflation rate. Next, in lower regime, exchange rate is associated positively to the economic growth and showing there is 0.41% increase in the economic growth in deflation or less persistent regime with a unit increase in the exchange rate. On reverse, in high regime, exchange rate is linked negatively and indicating that there is a 0.23% decrease in the exchange rate with a unit increase in the exchange rate. All the variables are in high and lower regime significant. In case, linear regression model, both inflation rate and exchange rate are insignificant and showing the negative association with economic growth in the economy.

At first, unit root tests are applied to see the stationarity of the data which show that all the variables are stationary of same order. Next, nonlinearity is detected in the inflation series which force to regress threshold cointegration. The null hypothesis of no cointegration is rejected and accepted the alternative of threshold cointegration in the model. The TVECM reveals that there is a different speed of adjustment parameter in different regime. It is quite valid and as according to the literature. In different phases economic growth reacting differently to the inflation and exchange rate in case of Pakistan. In lower regime, when there is deflation, our nonlinear model suggests that there is a weak sacrifice ration between economic growth and inflation. As, there is slow demand of money in deflation and it decreases in consumption which also lead to decrease in the economic growth. When there is increase in money supply, the demand of money also increases lead to increase in the consumption level which ultimately results in high prices at the same time more goods are produced in the economy. It shows that in deflation or less persistent regime in Pakistan there is a positive relationship between economic growth and inflation rate. On opposite to it, when there is high regime, which shows all the macro economic variables are at increasing trend it would lead to high consumptions due to which there is inflation in the economy goods are being unsold and thus results into to decrease in the economic growth of the country. Similar, condition is revealed by our nonlinear regression model in case of high persistent regime. Next, with exchange rate, economic growth is linked positively in above model, which shows at low level of prices there is depreciation in the economy which lead to increase our exports level and finally result into to increase the output of the economy and opposite to it in high persistent regime. Thus, the hypothesis of our study is estimated through the nonlinear models such to check the sacrifice ration in less persistent regime as well as in high persistent regimes. All these information is not available through linear regression model which only reveals the simple negative relationship throughout all the phases of the economy in case of Pakistan.


Roomer (1993) hypothesis has been tested as in case of less persistent regime and high persistent regime, using economic growth, inflation rate and exchange rate time series data comprises 1981q1 to 2016q4 in case of Pakistan. All the variables are found to be stationary at first order and nonlinearity detected in the inflation series with the threshold level of 7.1% for 2007q1. Threshold cointegrtion is accepted for the model and TVECM shows the different speed of adjustment parameters as in case of lower and higher regime. It shows that in case of depression the convergence among variables are slower as compare to the higher regime in case of Pakistan. The nonlinear regression model discriminated on threshold level in inflation is estimated which reveals that there is different association among economic growth and inflation in case of depression and boom in the economy. As, in case of depression, there is a positive relationship between economic growth and inflation rate but in boom there is a negative relationship between them and very similar conditions are exist for the exchange rate. As, in depression due to lower prices there is increase in the exports due to which there is net increase in the output of the economy. But in boom, due to higher prices, there is an appreciation due to which domestic goods become expensive and imports become cheaper which lead to decrease local production that ultimately decreases the output of the economy and imports.

Government and central bank should need to know at first the phases of the economic growth, that whether there is less persistent regime or high persistent regime in the economy. As, in case of less persistent regime government should lower down the taxes and increase the government expenditure to boost the economic growth on the other hand central bank should lower the interest rate and vice versa.


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Umer Hussain

M.Phil Econometrics

Pakistan Economic Research Institute (PERI)

S. M. Naeem Nawaz

Senior research fellow at Pakistan Economic Research Institute (PERI).

Pakistan Economic Research Institute (PERI)

Doi: 10.24312/paradigms110211
Table 1
Unit Root

Variable       Levels   First Difference
               c        c & t             c                  c & t

[y.sub.t]      -1.031   -1.221            -4.012 (*)         -4.412 (*)
[[pi].sub.t]   -1.941   -2.100            -3.911 (**)        -4.011 (*)
[er.sub.t]     -2.011   -1.066            -4.099 (*)         -3.099 (**)
                        Phillips-Perr     on
               Levels                     First Difference
[y.sub.t]      -2.031   -2.221            -3.912 (*)         -4.011 (*)
[[pi].sub.t]   -1.041   -0.100            -4.011 (**)        -4.000 (*)
[er.sub.t]     -1.011   -1.966            -3.099 (*)         -4.091 (**)

Note: (***), (**) and (*) which shows level of significane at 1%, 5%
and 10% respectively. c & t are constant and trend terms respectively.

Table 2
Threshold Level

Hypothesis                 F-test       T       C.I 95%

[H.sub.0]: No threshold    26.74 (**)   3.41   [1.855% to 4.367%]
[H.sub.0]: One threshold    4.18        2.01

Note: *, (**) and *** showing significane at 1%, 5% and 10% level. C.I
is confidence interval. Thr is threshold estimate.

Table 3
Threshold Cointegration

Dependent variable:                          [y.sub.t]

Test Statistic Value                         25.702
p-values (a)                                  0.008
p-values (b)                                  0.001
Threshold parameter ([??]) (c)                7.1
(*) Cointegrating value estimated ([??])      0.718, 0.211
intercept/Lag/ /trim/ ngridTh             0/1/0.1/no

Note: (a): Bootstrap value of Fixed regressor. (b), is the value of
bootstrap residuals. (c): is estimated threshold parameter. LM
consistent heteroscedastic test statistics:

Table 4

Variable                ECTt-1                      ECTt-1
                     [less than or equal to]  [greater than or equal to]
                       7.1%                        7.1%
                     Coef/std. E                  Coef/std. E

ECTt-1               -0.541 (**)                  -0.741 (**)
                     (0.012)                      (0.131)
Const                 0.019                        0.120 (**)
                     (0.110)                      (0.001)
[[DELTA][pi].sub.t]   0.191                       -0.355 (**)
                     (1.011)                      (0.022)
[[DELTA]er.sub.t]     0.141 (***)                 -0.211 (***)
                     (0.001)                      (0.101)
                      0.09                         0.2
Hetro(p-value)        0.56                         0.69
Adj-R2                0.59                         0.65

Note: (*), (**) and (***) are showing significance of coefficient at
1%, 5% and 10% level of significance. Further, in parenthesis standar
errors are given. Diagnostic tests are given as Autocorrealtion and
hetroscedasticity there p-value is below 0.05 and rejecting the null
hypothesis of auto and heteroscedasticity in the model estimated.
ECTt-1[less than or equal to]7.1% is showing regime one; less
persistent regime and the other one is high persistent regime.

Table 5
Nonlinear Regression Model

Vari                  L. Model                    Asymmetric Model
                     (excluding                   [[DELTA][pi].sub.t]
                     threshold)                  [less than or equal to]

Cons                  0.010         0.031
                     (0.001) (**)  (0.011) (**)
[[DELTA][pi].sub.t]  -0.001         0.313             (2.101)
                     (0.002)       (0.101) (**)       -0.169
[[DELTA]er.sub.t]    -0.163         0.411             (0.011) (**)
                     (0.180)       (0.121) (*)        -0.231
Obs                                58                 (0.012) (***)
[R.sup.2]             0.58          0.61
Auto                  0.09          0.44
Hetro                 0.12          0.81

Vari                    [[DELTA][pi].sub.t]
                    [greater than or equal to]7.1

Cons                         0.123
Obs                         84
[R.sup.2]                    0.56
Auto                         0.51
Hetro                        0.79

Notes: In above both model, dependent variable able is growth rate and
standard errors are given. And (*), (**), (***) are showing the
significance of the variables at 1%, 5% and 10% level of significance.
And the diagnostic test results are also given as
Auto-correlation/p-value and hetroscedasticity/p-value, through p-value
both null hypothesis are rejected. Regimes are divided by estimating
threshold value in inflation is 7.1% (2007Q1). Obs shows the total
observation in each regime. L. Model is linear regression model.
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Date:Jul 1, 2017
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