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Modeling and forecasting inflation in Rwanda (1995-2009)


par Ferdinand GAKUBA
Kigali Independent University - Degree in economics 2009
  

précédent sommaire

2.4 THE LONG RUN MODEL OF INFLATION

Various economists have attempted to empirically analyze the issues outlined in the
previous section. Earlier studies, such as Bourne and Persaud (1977) and Holder
and Worrell (1985), emphasized the role of structural influences and cost push

e found that monetary disequilibrium and exchange

xplaining the behaviour of prices in the Jamaican economy46. The link between the money stock and inflation occurs via a monetary transmission process whereby the amount of money economic agents desire to hold is less than the available money stock. Assuming a stable demand for money, this serves to reduce the value of money (in terms of goods) thus increasing the price level.

We estimated a model similar to the Harberger model using ordinary least squares. The results using quarterly are;

DLIPC = 0.104*LEXCESS_MONEY(-1) + 0.062*LMARKUP + 0.099*DLM2 - 0.0327*DLOPC + 0.20*DLPIBC + 0.21*DLTCUSD(-1) - 0.0034*DLTDC(-1) + 0.184*DLULC - 0.0067*DLIP

With R-squared= 0.42, Schwarz criterio=-4,38, F-statist= 1,23 ,DW= 1.25, Chow = 1.11, Normality test= 1,32 and ARCH= 0.38 sigma= 0.96

We estimated a general model in which we regressed Dlcpi (difference in logarithm of consumer price index) on the above mentioned long run (and structural) relationships markup,Lexcess_moneyt-1, and Dlpibc, and short run dynamics Dltcusd(-1), Dlulc, Dlip, DLm2 and DLtdc Our sample goes from 1995Q1 until 2009Q2 and for that sample period our unrestricted general model yields sigma = 0.96 percent for 9 regressors and 56 observations (Schwarz criterion = -4.38). Next step was eliminating insignificant terms from the model by allowing the log variables stationary

The procedure followed is general to simple approach

LIPC = 0.016*LIP(-1) + 0.81*LIPC(-1) + 0.121*LM2 - 0.047*LOPC - 0.025*LPIBC(- 1) + 0.020*DLTDC(-1) _ 0.077*DLULC(-1) + 0.406*DLTCUSD

46 Wayne Robinson (1998) : Forecasting inflation using VAR analysis, Bank of Jamaica

26 F-statistic= 1060,04 , Jarque- bera= 9.14,ARCH

est with also high probability over 5%. The model proved its adequacy in terms of various diagnostic tests and also it encompasses the unrestricted general model. For more information see appendix 6

The contemporaneous money stock, import price, oil price, and exchange rate had expected signs and were very significant. The results suggest that the money supply, import prices, mainly the oil prices fluctuations and exchange rate changes had the largest impact on price changes. Using the quarterly data the model derived is;

LIPC = 0.029 + 0.016* LIP t-1 + 0.81*LIPC t-1 + 0.121*LM2 -0.047* LOPC- 0.026* LPIBC t-1 +0.020* DLTDC t-1 - 0.077*DLULC t-1 +0.40*DLTCUSD

Table 7: Results for long run inflation model

Dependent Variable: LIPC

Method: Least Squares

Date: 11/06/09 Time: 09:41

Sample (adjusted): 1995Q3 2009Q2

Included observations: 56 after adjustments

Variable

Coefficien

t

Std. Error

t-Statistic

Prob.

LIP(-1)

0.016072

0.008212

1.957190

0.0563

LIPC(-1)

0.815484

0.079861

10.21133

0.0000

LM2

0.121700

0.042441

2.867535

0.0062

LOPC

-0.047478

0.018775

-2.528729

0.0149

LPIBC(-1)

-0.026858

0.037109

-0.723760

0.4728

DLTDC(-1)

0.020468

0.025706

0.796245

0.4299

DLULC(-1)

-0.077711

0.061559

-1.262391

0.2130

DLTCUSD

0.406268

0.137711

2.950146

0.0049

C

0.029796

1.020664

0.029193

0.9768

R-squared

0.994488

Mean dependent var

 

0.036243

Adjusted R-squared

0.993550

S.D. dependent var

 

0.267714

S.E. of regression

0.021500

Akaike info criterion

 

-4.695275

Sum squared resid 0.021726

Schwarz criterion

 

-4.369772

Log likelihood

140.4677

Hannan-Quinn criter.

 

-4.569078

F-statistic

1060.045

Durbin-Watson stat

 

1.268751

Prob(F-statistic)

0.000000

 
 
 

ighly influenced by lagged inflation, money supply

ese results also highlight the significant role of oil prices and import prices, starting from the hypothesis of the Quantity Theory, estimated the relationship between money supply and prices in Rwanda between 1995 and 2009. The changes in prices were examined as a function of changes in the money supply (M2), previous price changes and changes in the exchange rate. Using quarterly data, the estimated model most preferred was;

LIPC = 0.029 + 0.016* LIP t-1 + 0.81*LIPC t-1 + 0.121*LM2 -0.047* LOPC- 0.026* LPIBC t-1 +0.020* DLTDC t-1 - 0.077*DLULC t-1 +0.40*DLTCUSD

2.4.1 Interpretation of coefficients

R- squared= 99.44% this mean that the 99.44 % fluctuations of prices are explained by last inflation, money supply, exchange rate, import price and oil shocks.

Most coefficients are statistically significant only the coefficients of LPIBC, DLTDC, and DLULC have a high probabilities the reason for that is because the PIBC was estimated for quarterly the ULC was calculated using NSSF data. This could lead to estimators bias.

2.4.2 Classical Tests

-The T- Student of LIP t-1, LIPC t-1, LM2, LOPC and DLTCUSD have the significant influence on inflation.

-Homocedasticity test of correlation of errors

Ho= the model is homocedastic H1= the model is heteroscedastic

Table 8: Heteroskedasticity test

White heteroskedasticity test: No cross terms

F- statistic

1.093951

Probability

0.3842

Obs*R- squared

8.79061

Probability

0.3603

White heteroskedasticity test: cross terms

F- statistic

2.034542

Probability

0.0508

Obs*R- squared

44.46531

Probability

0.1572

With the two option test above we accept the first assumption that there is homoskedasticity because the probabilities are high than 5%

H1= errors correlated

We dispose here the sample n= 56 observations. The number of real explanatory variables is K=8 on Durbin Watson table at 5% level of freedom; Dinf=1,39 and Dsup= 1,51 and our DW calculated is =1, 26 this mean that there is presumably a positive autocorrelation of errors.

We correct the autocorrelation with the Cochrane Orcutt method by adding the inverted AR root

The DW find after new estimation is= 1.68 and K have been 9 so Dinf =1,32 and Dsup= 1.58 mean that we presumably resolve the autocorrelation problems by Cochrone Orcutt method

Ho is accepted no correlation among errors.

Test of Ramsey RESET

The assumptions for this test are as follows; Ho= the specification of model is good

H1= the model is badly specified

Table 9: Ramsey RESET test Ramsey RESET TEST

F- statistic

1.3055

Probability

0.2702

Log likelihood ratio

13.3559

Probability

0.1002

These two probability show that the model specification is good because the probalitity are higher than 5% mean that we accept Ho.

Figure 5: CUSUM Test

If the cusum curve is out of corridor means that the coefficients of the models are unstable and curve does not leave the corridor mean that the coefficients are stable.

CUS UM 5% S ignific anc e

0

- 10

- 20

- 30

98 99 00 01 02 03 04 05 06 07 08

Figure 6: CUSUM squared test

This test allow to detect the punctual instability

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

- 0.2

- 0.4

 
 

98 99 00 01 02 03 04 05 06 07 08

 

CUSUM of Squares 5% Significance

 

With the cusum squared line which lie between the corridor line mean that the model is stable in all coefficients

Partial conclusion

From this we conclude that rise of price is influenced by changes in the money supply, but not directly as the Quantity Theory suggests. Monetary changes affect inflation indirectly because of the prevalence of mark-up pricing. This also provides the channel for the impact of exchange rate adjustments (i.e. changes in the exchange rate affect variable cost) and lagged prices.

Here taking consideration of Rwanda context the inflation appears to be driven by
both foreign and domestic factors in a manner consistent with conventional
theoretical models and the Rwanda inflation is not only the monetary phenomenon.

hat monetary policy has a lag effect of `at least' two

e perpetuated by the nature of the stabilization process, the structure of the economy, the production function, and other institutional factors. Other factors such as the oil pricing mechanism, import level and exchange rate, which are captured in the own innovations of the CPI and exchange rate are also very significant and create very strong inertial tendencies. Stabilization policies must therefore be cognizant of these influences that frustrate the stabilization process.

 

TION OF RWANDA USING ECM MODEL

 

In the second chapter we have seen the inflation long run model but Monetary policy-makers face a difficult task when evaluating the current state of the economy and deciding what actions are needed to achieve their objectives, such as keeping inflation within a given range. Because long and variable lags exist between a monetary policy action and its effects on economic variables, policy-makers need a way to assess whether their actions are having, or indeed will have, the desired effect.

The model used in this paper is similar to Hendry's original model in that it estimates a unique and stable long-run cointegrating vector between quarterly data for cpi, nomimal output, the M2, exchange rate, import price , oil price index, unite labor cost and a short-term interest rate.

3.2. Error correction model of inflation

The error correction model represents one of remarkable property which had been demonstrated by Granger (1983). The whole variables cointegrated could set in form of error correction model where all variable are stationary and the coefficients could be estimated using the classical econometric approach without correlation risks.

Here we propose two types of error collection models;

3.2.1 The Hendry model

The model based on an error correction mechanism first introduced by Sergan (1964) and popularized by Hendry in numerous papers has enjoyed a revival in popularity in empirical macroeconomics research is expressed as47;

DLIPC =130+ 131*DLIP(-1) +132*DLIPC(-1) + 133*DLM2 +134*DLOPC + 135*DLPIBC(-1) +136*DLTDC(-1) + 137*DLULC(-1) +138*DLTCUSD +139*LIPC(-2)+ 1310*LM2(-1)+ 1311*LOPC(-1)+1312*LPIBC(-2)+ 1313* DLTDC(-2)+ 1314*DLULC(-2)+ 15*DLTCUSD(- 1)+ Vt

and Vt is error terms ,the coefficients 130 138

ic and the â10 â15 characterized the long run

equilibrium the 139 is the error correction coefficient which could be under one unity and has a negative sign. The error correction coefficient indicate the speed of adjustment of explained variable IPC by returned on equilibrium in long run follow the shock. The 130 represent the model constant.

Using the OLS the resultants coefficients are as follow;

DLIPC = -0.63 + 0.0089*DLIP(-1) + 0.207*DLIPC(-1) + 0.0348*DLM2 - 0.015*DLOPC + 0.0277*DLPIBC(-1) - 0.0116*DLTDC(-1) - 0.018*DLULC(-1) + 0.183*DLTCUSD +0.010*LIP(-2) -0.224*LIPC(-2) +0.111*LM2(-1) _0.027*LOPC(-1) --0.10*DLPIBC(-2) - 0.0279*DLTDC(-2) --0.082*DLULC(-2) - 0.21*DLTCUSD(-1)

The interpretation of 139 coefficients (restoring force to balance) has a negative sign as predicted. We find that the coefficients associated with the restoring force is - 0.224 and clearly significant different from zero at level of confidence equal to 5% (his T- student is greater than 1.96 in absolute value) mean that there is therefore a mechanism for error correction; in long run the disequilibrium between IPC and all others explanatory variables is compensated so that all series have similar evolution. B9 represent the speed at which any imbalance between desired levels and the effective level of inflation is eliminated in the year following shock. We get adjusted 22.4% of the imbalance between desired and actual level of inflation.

3.2.1.1 Long run and short run elasticity

The short run elasticity is represented by 131 138 coefficients; if the level of

import increase by 10% then the inflation rise by 0.089%, if the money supply increase by 10% the inflation rise by 0.348%, if the oil price increase by 10% the rise in prices will be 0.15%,if the GDP rise by 10% the prices level reduce the 0.277%, if interest rate rise the one point the prices level rise the 0.116% and if the Rwanda currency appreciated the 10% compare to foreign currency the prices level will rises the 1.83%.

47 James P. Le sage@ (1990) ; A Comparison of the Forecasting Ability of ECM and VAR Models, page 23

ted by 1310 1315 coefficients;

0.111

Money supply elasticity = = = 0.495

B9 0.224

this means that in long run if the money supply rise by 10 % the level of price will rise by 4.95%.

1315 0.210

Exchange rate elasticity = = = 0.937

B9 0.224

If the Rwanda currency appreciate by 10% the level of price will rises by 9.37% in whole economy.

B13 0.0279

Interest rate elasticity in long run= = = 0.124

B9 0.224

This mean that as soon as the interest rate increases by one point the level of price

rise by 1.24% in long run.

B14 0.082

Wage elasticity = = = 0.366

B9 0.224

The wage is one of the significant variables which cause the price fluctuations in

economy as soon as the nominal rise the 10% the price level pass at 3.66% in long run.

B12 0.027

Oil price elasticity = = = 0.12

139 0.224

Oil shock have also impact on price fluctuations in long run, the rise of 10% on oil

prices rise 1.2% of prices level in general.

3.2.1.2 Significativity of error correction model

T- student all variables are not significant

F- statistic show that is small mean that the model is not fit as well

R- squared is also small which mean than model is not good.

Jarque - Bera test

With the probability equal to 0.000545 which is lower than 5% allow as to reject Ho state than the error distribution doesn't follow the normal law of distribution.

 
 

F- statistic

0.576303

probability

0.8819

obs*R-squared

10.73988

probability

0.8252

 

The errors of ECM are homoskedatic

ARCH test

Ho= errors are homoskedastic

H1= errors are heteroskedastic

The errors are homoskastic if the probability is higher than 5% and the errors are heteroskedastic if the probability is lesser than 5%.

Table 11: ARCH test

ARCH heteroskedasticity Test

F- statistic

0.001992

probability

0.9646

obs*R-squared

0.002069

probability

0.9637

 

The errors are homoskedastic

Breusch- Godfrey test

Ho= errors are not correlated

H1= errors are correlated

We accept the no correlation assumption when the probability is higher than 5%.

Table 12: Breusch -Godfrey Test

Breusch- Godfrey Serial Correlation test

F- statistic

6.770304

probability

0.0030

obs*R-squared

14.43214

probability

0.0007

 

The errors of ECM are correlated the estimated coefficients get are bias.

Ramsey RESET test

F- statistic

3.960093

probability

0.0090

likelihood

19.95602

probability

0.0005

 

With the two probability above are less than 5% mean that the model is badly specified

Figure 7: CUSUM stability test (Brown, Durbin, Ewans)

20 15 10 5 0 -5

- 10

- 15

- 20

 
 
 

CUSUM5% Significance

The Cusum curve lie between corridor lines mean that the ECM is structurally stable

Figure 8: CUSUM squared stability test

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

- 0.2

- 0.4

 
 

99 00 01 02 03 04 05 06 07 08

CUSUM of Squares 5% Significance

e ECM is instable the 2005-2006 is instable period.

e unadequancy and is instable at certain period so due to lack of certain dummies variables which could be included to correct it, so we didn,t have during this estimation so we propose another ECM recommended by Engle- Granger with two-step.

3.2.2 The ECM with Engle - Granger

The model of Granger enables the long-run equilibrium relationship and short-run dynamics to be estimated simultaneously particularly for finites samples, where ignoring dynamics when estimating the long run parameters can lead to substantial bias48. One of the advantages of this specification is that it isolates the speed of adjustment parameter C(9), which indicates how quickly the system returns to equilibrium after a random shock. The significance of the error correction coefficient is also a test for cointegration. Kremers, Ericsson and Dolado (1992) have shown this test to be more powerful than the Dickey-Fuller test applied to the residuals of a static long-run relationship. Another reparameterisation, the Bewley (1979) transformation, isolates the long-run or equilibrium parameters and provides t-statistics on those parameters. Inder (1991) shows these approximately normally distributed t-statistics are less biased than the Phillips-Hansen adjusted t-statistics The model is expressed as follow;

DLIPC = C(1) + C(2)*LEXCESS_MONEY + C(3)*DLMARKUP + C(4)*RESID01(-1) + C(5)*LTDC(-1) + C(6)*DLM2(-2) + C(7)*LOPC(-1) + C(8)*DLTCUSD(-1) + C(9)*DLPIBC(-1) + C(10)*LULC(-1)

The results for estimated model are as follow;

DLIPC = -0.0199- 0.173*LEXCESS_MONEY + 0.038*DLMARKUP - 0.319*RESID01 (-1) - 0.030*LTDC (-1) + 0.067*DLM2 (-2) + 0.033*LOPC (-1) + 0.062*DLTCUSD (-1) + 0.165*DLPIBC (-1) + 0.019*LULC (-1)

48 Banerjee et al. (1993) and Inder (1994) show that substantial biases in static OLS estimates of the cointegration parameters can exist, particularly in finite samples, and the unrestricted error correction models can produce superior estimates of the cointegrating vector.

e level of priced in Rwanda is negatively related to
elated to mark up and this is a short run correlation

between variables

3.2.2.1 Long-run elasticity and short run elasticity

The short run elasticity is represented by 134 139 coefficients; if the level of

money supply grow by 10% then the inflation rise by 0.067%, if the exchange rate fluctuate by 10% the inflation rise by 0.030%, If the oil price increase by 10% the inflation rise by 0.033%, if the salaries rise by 10% the inflation rise by 0.019, if the interest rate grow by one point(mean by 100%) the inflation also rise by 0.062% and if the GDP grow by 10% the inflation decrease by 0.165% ,this allow as by making sure that the inflation is also originated from foreign fluctuations.

The long run elasticity is represented by 131 132 coefficients;

131 0.173

Money supply elasticity = = = 0.542

134 0.319

this means that in long run if the money supply rise by 10 % the level of price will rise by 5.42%.

131 0.038

Exchange rate elasticity = = = 0.119

134 0.319

If the Rwanda if foreign price rise as results of import changes by 10% the level of price will rises by 1.19 in whole economy.

Before turning to the results, it is necessary to consider the statistical properties of the model. The model was tested for normality, serial correlation, autoregressive conditional heteroskedasticity, heteroskedasticity, specification error and stability. The results, reported in Table 5, suggest the model is well specified. The diagnostics indicate that the residuals are normally distributed, homoskedastic and serially uncorrelated and the parameters appear to be stable.

test

 

values

Probability

Jarque-Bera

q 2-statistic

2.7080

0.0980

Breusch-Godfrey

F-statistic

0.213537

0.9911

Correlation LM test

q 2-statistic

2.245793

0.9870

ARCH LM test

F- statistic

0.008989

0.9248

 

q 2-statistic

0.009327

0.9231

white

F- statistic

0.140097

0.9982

 

q 2-statistic

1.494021

0.9972

 

Chow breakpoint Test(

F- statistic

1.801709

0.0958

mid sample)

L-R-statistic

18.01709

0.0547

Chow Forecast test

F- statistic

0.638972

0.8506

(1995-2009)

L-R-statistic

57.86670

0.0065

Ramsey RESET test

F- statistic

4.049290

0.0502

 

L-R-statistic

4.825140

0.0280

Notes :**(*) denotes significance at the one (five) per cent levels. No terms were

significant at these levels. LR is a likelihood ratio statistic.

Figure 9: Residuals test

.04 .03 .02 .01 .00

-.01

-.02

-.03

-.04

 

98 99 00 01 02 03 04 05 06 07 08

Rec urs ive Residuals #177; 2 S .E .

CUS UM 5% Signific anc e

Granger ECM

10

0

-10

20

98 99 00

01 02 03 04 05 06 07 08

Figure 11: Cusum of squared test for Engle- Granger ECM

1. 4 1. 2 1. 0 0. 8 0. 6 0. 4 0. 2 0. 0 -0. 2 -0. 4

 

98 99 00 01 02 03 04 05 06 07 08

CUSUM of S quares 5% Signific ance

From the above test results the model proves its adequacy in term of coefficients and p-values. The results provide strong support for the conventional ECM model as a description of inflationary processes in Rwanda. They are consistent with the conventional theory, and with the findings of many overseas studies. They are also consistent with our understanding of the institutional structure of the domestic economy. The results suggest that about three quarters of the long-run movement in prices in Rwanda has been underpinned by import prices; about one quarter has been driven by domestic labour costs. The small coefficient on the error correction term points to protracted periods of disequilibrium for long run and drawn out adjustment processes, particularly in respect of changes to unit labour costs. In the interim, domestic demand conditions play an important role. Many indicators are using to evaluate the quality of the model proved to be used as prediction model the more used are MAPE (mean absolute percentage error), and the Inequality coefficient of THEIL, the model with small MAPE is judged

sion and Theil comprised between 0 and 1 indicate r instance (see figure 13)

Figure 12: Inflation forecasting criteria

Forecast: DLIPCF

Actual: DLIPC

Forecast sample: 1995Q1 2009Q2 Adjusted sample: 1995Q4 2009Q2 Included observations: 55

Root Mean Squared Error

0.012813

Mean Absolute Error

0.009682

Mean Abs. Percent Error

93.46577

Theil Inequality Coefficient

0.250929

Bias Proportion

0.000000

Variance Proportion

0.119487

Covariance Proportion

0.880513

.10 .08 .06 .04 .02 .00 -.02 -.04 -.06

 
 

1996 1998 2000 2002 2004 2006 2008

 
 

DLIPCF #177; 2 S.E.

 

The two models seen above have the same characteristics and don't allow us to make the best forecast because the mean absolute percentage error is high in two equations above see Figure 6. Instead of using one of them we extend our understanding by apply the VECM as theoretical suggest. The main reason for estimating a VECM system of equations is because arguments call for the lag between cause and effect to be shorter than the forecast horizon. Forecast the causal variables are needed and estimating VECM will automatically provide them. We initially estimate equation in level not in first differences because it's often possible to find a group of variables that is stationary even thought the individual variable are not. Such a group is cointegrating vector, if the values of two variables tend to move together over time so that their values are always in the some ratio to each other, then the variables are cointegrated. This is the desirable long run relationship between a causal and dependent variables. The value of the causal predicts the value of the dependent variable but in particular time the prediction is not

er be large. An article that should become classical 1994)49.

3.3. Forecasting inflation using VECM

An error correction model contains one or more long run cointegration relationship as seen above but using the VECM will while not necessary to go back to cointegration concepts is helpful in making the connection between a VAR with variables in levels the error correction form and a VECM with variable in differences. From a theory standpoint the parameters of the system will be estimated consistently and even if the true model is in differences, hypothesis tests based on an equation in levels will have the same distribution as if the correct model had been used.

3.3.1 Why a VECM?

Because any exercise in empirical macroeconomic must recognize the conclusion drawn from times series analyses of macroeconomic data, and utilize specifications that are consistent with these results. Such analyses starting with the classic study of Nelson and Plosser (1982), consistently have demonstrated that macroeconomic time series data likely include a component generated by permanent or nearly permanent shocks. Such data series are said to be integrated, difference stationary, or to contain unit roots. On the other hand, economic theories suggest that some economic variables will not drift independently of each other forever, but ultimately the difference or ratio of such variables will revert to a mean or a time trend50.

Granger defined variables that are individually driven by permanent shocks (integrated), but for which there are weighted sums (linear combinations) that are mean reverting (driven only by transitory shocks), as cointegrated variables. He then demonstrated in the Granger Representation Theorem (Engle and Granger, 1987; Johansen, 1991) that variables, individually driven by permanent shocks, are

49 Murray, M. P(1994), «A drunk and her dog: An illustration of cointegration and error correction,» American Statistician, 48, 37-39.

50 Klein, L.B. and B. F. Kosobud (1961), «Some Econometrics of Growth: Great Batios of Economics», Quarterly Journal of Economics, 75:173-98.

xists a Vector Error Correction representation of the

3.3.2. Forecasting performance

While the cointegrating vectors determine the steady-state behavior of the variables in the Vector Error Correction Model, the dynamic responses of each of the variables to the underlying permanent and transitory shocks are completely determined by the sample data without any restriction.

Forecasting performance may be gauged in a number of different ways. Papers by Clements and Hendry (1993) and Hoffman and Rasche (1996b) employ measures of system performance, while Clements and Hendry (1993) and Christofferson and Diebold (1996) argue that conventional RMSE criterion may not capture some of the advantages of long-run information into the system. The basic conclusion of this body of literature is that incorporating cointegration may improve forecast performance, but improvement need not show up only at longer horizons as predicted originally by Engle and Yoo (1987). The advantage presumably accrues from the addition of error correction terms in VECM representations. Christofferson and Diebold (1996) contend that conventional RMSE criterion will not capture this forecast advantage at long forecast horizons simply because the importance of the error correction term diminishes with increases in the forecast horizon. For the exercise we have in mind, the relevant issue is forecast performance for a subset of the variables in our system, at various horizons, and the most relevant measure of that performance is the standard mean squared error criterion. We employ RMSFE as a criterion while recognizing that it may not capture all the advantages that the long-term information has to offer.

The results for VECM are in the following table 16

The results show that the fluctuations of price level are positively related to previews price level and the mark up but negatively related to nominal GDP, exchange rate, the interest rate and the excess money supply.

51 Engle, R.F. and C.W.J. Granger (1987), «Cointegration and Error Correction: Representation, Estimation, and Testing» Econometrica, 55:251-276.

esponse of shocks on errors for all variables. For

al to 2 standard deviation of errors. The temporal Horizon for response is set at 10 quarterly, this horizon represent the time needed in which the variable recover the long run level.

Figure 13: Response function of variables on LIPC

Response of LIPC to Cholesky Response of LMARKUP to Cholesky Response of LEXCESS_MONEY to Cholesky

One S.D. Innovations One S.D. Innovations One S.D. Innovations

Response of LM2 to Cholesky

LPIBC LTCUSD

One S.D. Innovations

Response of LPIBC to Cholesky
One S.D. Innovations

LIPC LMARKUP LEXCESS_MONEY

LM2

Response of LTCUSD to Cholesky
One S.D. Innovations

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

.08

.06

.04

.02

.00

-.02

-.04

.06

.04

.02

.00

-.02

-.04

-.06

.10

.03

.02

.01

.00

-.01

LIPC LMARKUP LEXCESS_MONEY

LM2 LPIBC LTCUSD

LIPC LMARKUP LEXCESS_MONE

LM2 LPIBC LTCUSD

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

.06

.05

.04

.03

.02

.01

.00

-.01

.06

.04

.02

.00

-.02

-.04

-.01

-.02

-.03

-.04

.03

.02

.01

.00

1 2 3 4 5 6 7 8 9 10

LIPC LMARKUP LEXCESS_MONE

LM2 LPIBC LTCUSD

LIPC LMARKUP LEXCESS_MONEY

LM2 LPIBC LTCUSD

LIPC LMARKUP LEXCESS_MONEY

LM2 LPIBC LTCUSD

The shock is positive on LIPC generated by the negative effect of money supply, exchange rate, interest rate, and nominal GDP as seen above (seen figure 14) then the shock is negative to LIPC generated to positive effect of previews price level and mark up the whole fructuation on any variable have a significant impart on price level the figure above shows how change is made by mark up, excess money supply, money demand and GDP every period.

ition

The aim of decomposition of the variance of errors of prevision is to calculate for each innovation have as contribution on the errors variance expressed in percentage, when the innovation explain the big part in variance of error we conclude that the considered economy is sensible on shock which affect the series seen the appendix 5. The decomposition of the variance show that the variance of error prevision of the LIPC is due to 80% on its own innovation, 1% of the mark up innovation,10% of excess money supply innovation, 7% of M2 innovation ,1% of GDP innovation and 2% of exchange rate innovation.

The variance of error of the others variable is represented on the appendix 5 all of them show the accuracy.

.3.3.2.2 Forecast results

The forecast was produced using the data available through the first quarter of 1995. Table 16 provides the numeric forecasts both quarter-by-quarter and on an annual basis. The annual values for the levels of real GDP, nominal M2, and the two interest rate series are four-quarter averages. The annual values for the inflation rates and growth rates are measured on a fourth-quarter to fourth-quarter basis. Real growth for the third quarter of 2009 is forecast to be very slow, indeed to decline from the rate of 2010.1. In retrospect we know that this was a really large forecast error, since the third quarter of 2009 came in exceptionally strong. The forecast is for continued slow real growth (< 1 percent per quarter) though the end of 2010. This certainly will be an underestimate of real growth for 2009, and appears at this time (January, 2010) to be an underestimate of real growth for 2010.

In addition, the forecast did not catch the large increase in long rates and the accompanying increase in the slope of the yield curve that began in the last part of the 2010:1 and continued through third quarterly. In comparison, the money supply (M2) is projected to remain essentially constant through 2009 (around 6.10 percent), not far from the actual experience. Thus the long-term rate forecast errors in 2010 are attributable to errors in the implicit short-run term structure relationship. Taken literally, the model is forecasting a reduction in the exchange rate target to around

10. Finally, the model projects no change in the

o inflation rates (at roughly 2.5 percent per annum in terms of the CPI inflation rate) through the end of 2010. Hence the model is not predicting any increase in the long-run rate of inflation over this period.

Table 15: Prevision results

period

ipc

Lmarkup

Lexcess money

Lm2

lpibc

ltcusd

2009 q3

0.648560

0.184016

0.200919

6.078984

29.08976

6.370031

2009q4

0.538254

0.016532

0.262129

6.431286

29.28959

6.331708

2010q1

0.564918

0.297554

0.464941

6.664537

28.93297

6.303760

2010q2

0.621301

0.263436

0.607381

6.931436

29.97537

6.221759

2010q3

0.886635

0.114588

0.73921

6.506427

29.03635

6.295411

2010q4

0.701047

0.234921

0.499361

6.863611

29.95079

6.204930

The table15 above show the prevision of future variables and demonstrate how change will have the price level due to change of other variables here the variable are in percentage change. For example to comparing two period in 2nd quarterly of 2009 inflation was 185.30 expressed as CPI and 3rd quarterly the price level is 185.75 mean the evolution of 0.65% per/quarterly, the price level in 4th quarterly will be 186.04 mean evolution of 0.54%, the price level of 1st quarterly 2010 will be 186.38 mean evolution of 0.62% per quarterly, in the 2nd quarterly of 2010 the price level will be 186.86 mean evolution of 0.89% per period and the 3rd quarterly price level will be 187.23 mean the evolution of 0.7% per period. These fluctuations are results of demand shocks on one hand and the supply shocks on other.

model for forecasting because there are weighted sums (linear combinations) that are mean reverting (driven only by transitory shocks), as cointegrated variables and each variables is individually driven by permanent shocks, are cointegrated if and only if there exists a Vector Error Correction representation of the data series Nelson and Plosser (1982). The VECM has predicted inflation reasonably well over history and still appears to be a good forecasting model, especially in light of modifications like using adjusted, identifying policy shocks, and deriving probabilities for inflation outcomes.

The Rwanda inflation is driven in great part by the increase in money supply as it have been seen above but the excess money supply doesn't have the direct impact on price level its act through the exchange rate, and interest rate channel. These results provide useful insights into the behavior of inflation in Rwanda and the role of the National Bank of Rwanda in its determination. Inflation appears to be driven by both foreign and domestic factors in a manner consistent with conventional theoretical models and our understanding of the institutional structure of Rwanda economy. The results suggest that maintaining low inflation in coming years will depend largely on low level of interest rate, import, oil products in partner's countries and moderate growth of money supply and domestic unit labour costs. We have analyzed the inflation in two ways; first by the demand side as demand push inflation and on other hands by the supply side as cost push inflation.

1. On demand side for its part, the Rwanda national bank will need to maintain monetary conditions consistent with low inflation and low inflation expectations. In part, this is done through the pegged exchange rate, which acts as a nominal anchor for the economy. In this paper we wanted to analyze the driving forces of inflation process in Rwanda as a transition economy. First we derived, according to the theory, long-run sectoral relationship affecting inflation (markup, excess money, nominal exchange rate and GDP). Then we estimated short-run structural inflation function by imposing above mentioned long-run relationships together with various short-run variables which might contribute to explaining the inflation process in Rwanda, we conclude that inflation in Rwanda rise when money supply increase this by the increase in aggregate demande due also by the increased private and government spending.

2. On the supply side the mark up is seen to be significant because the increase of oil price, and import price increase at the sime time and immediately the price of domestic goods. As economic policy which is liberalized the producers to whom the oil is part of their cost could then pass (this change in price level) on to consumers in the form of increased price. In the derive quarterly ahead model of inflation, we found that all long run relationships except GDP significantly influence quarter-on-quarter

monetary variables are found to be important in

ar as narrow monetary aggregate is concerned, one can notice that the magnitude of its influence on consumer price inflation is quite marginal.

For forecast purpose we designed ECM model first by Hendry equations is not a perfect one, secondly by Engle Granger which proven instable by forecast criteria this is because of a high number of parameters estimated, while the definability of some equations is relatively high (a low relative coefficient of determination). However, they may be applied to the forecasting of inflation because the mean square errors of the forecast conform to the selected minimum criteria (1 or 5% should be mentioned). One of the main drawbacks of the VECM model, where the VAR methodology is used, is the fact that each time additional observations appear and the model is estimated as new equation of each group may be complemented (or reduced) by different variables. However, given stable parameters, the accuracy of results is not damaged. Nevertheless, the designing of a structural VAR or multi-equation econometric model for the forecasting of the CPI should probably be considered in future.

The VECM has predicted inflation reasonably well over history and still appears to be a good forecasting model, especially in light of modifications like using adjusted, identifying policy shocks, and deriving probabilities for inflation outcomes. Forecasts from the VECM can augment the information coming from other models used at the Bank. They can provide alternative views of what could happen in the economy and give some information about the «balance of risks.» Multiple models could be especially helpful to policy-makers during times of extreme uncertainty and/or structural shifts, but even in relatively stable times, advice from different models helps to balance risks about the outlook for the future.

According to the accuracy of the forecast calculated by the VECM model, this model could be suggested as a tool applied by the economists-analysts in the decision making procedure.

Some caveats are in order. First, the approach in this paper has been to evaluate forecasting performance using a simulated out of sample methodology. This methodology provides a degree of protection against overfitting and detects model instability. However, because a large number of forecasts were used, some over fitting bias nonetheless remains. This suggests that some of the best-performing forecasts produced using individual economic indicators might deteriorate as one move beyond the end of our sample. That is why the model used here could be improved in the future because;

- Coefficients of predictors can change over time

- The number of predictors can be large

- The relevant model for forecasting can potentially change over time

- The variables can also change over time

Second, we have considered only linear models. To the extent that the relation between inflation and some of the candidate variables is nonlinear, these results understate the forecasting improvements that might be obtained. Moreover, with few exceptions, incorporating other variables could be necessary to improve upon the short run forecasts and must be considered by National Bank agent in their future research.

1. Adam Smith. (1776), «Wealth of nations»

2. Ando, A. and F. Modigliani (1963), «The `Life-Cycle' Hypothesis of Saving: aggregate Implications and Tests», American Economic Review.

3. Banerjee, A., J. Dolado, J.W. Galbraith and D.H. Hendry (1993). Co-integration, Error-Correction, and the Econometric Analysis of Non-Stationary Data, Oxford University Press, Oxford.

4. Bewley, R.A. (1979). The Direct Estimation of the Equilibrium Response in a Linear Dynamic Model, Economic Letters.

5. Bomhoff, E.J. (1991). Inflation in Western Europe, paper prepared for the Fifth International Conference sponsored by the Institute for Monetary and Economic Studies, Bank of Japan, October.

6. Campbell, J.Y. and R.J. Shiller (1987),»Cointegration and tests of Present value Models» Journal of political Economy, 95

7. Duesenberry, J. (1950). The Mechanics of Inflation, Review of Economics and Statistics, 32 (2).

8. Edwards, Sebastian. (2002), «The great exchange rate debate after Argentina», The North American Journal of Economics and finance, Volume 13, Issue 3

9. Engle, R.F. and C.W.J. Granger (1987), «Cointegration and Error Correction: Representation, Estimation, and Testing» Econometrica, 55

10. Gary K. and Dimitris K, (2009), Forecasting Inflation Using Dynamic Model Averaging University of Strathclyde, June 2009

Principle of forecasting; A handbook for

A

12. Kiley, Michel J. (2008). «Estimating the common trend rate of inflation for consumer prices and consumer prices excluding food and energy prices». Federal reserve Board.

13. Klein, L.R. and R. F. Kosobud (1961), «Some Econometrics of Growth: Great Ratios of Economics», Quarterly Journal of Economics, 75

14. Kremers, J.J.M., N.R. Ericsson and J.J. Dolado (1992). The Power of Cointegration Tests, Oxford Bulletin of Economics and Statistics, 54 (3)

15. Mankiw, N.G.(1998) «Economic principles». New York: university Publishers,de Croisillon de Harcourt.

16. Mishkin, F. S. (1992), Is the Fisher effect for real: A reexamination of the relationship between Inflation and Interest rates, Journal of Monetary Economics 30

17. Murray, M. P(1994), «A drunk and her dog: An illustration of cointegration and error correction,» American Statistician, 48

18. Paul H. Walgenbach, Norman E. Dittrich and Ernest I. Hanson,(1973).»The Measuring Unit principle»

19. Payne, James E., (2002), «Inflationary Dynamics of a Transition Economy: the Croatian Experience», Journal of Policy Modeling, 24(3),

20. Sargent, Thomas J. (1986), «Rational Expectations and Inflation», New York; Harper and Row.

21. Selgin, G.A,(1989) «The analytical Framework of real bill Doctrine» Journal of institutional and theoretical economics, Volume 145,

22. Stock J. and Watson M. (1999), Forecasting inflation. Journal of Monetary Economics 44,

2001) : Research methodology course for the es , 2nd edition , Paris, Durod

Working Papers

1. Bandawe, H.P. (1997). Causes of Inflation in Fiji: Experience during 1979-1994, working Paper, Reserve Bank of Fiji.

2. Brower, G. and N.R. Ericsson (1995). Modeling Inflation in Australia, Reserve Bank of Australia, Research discussion Paper No. 9510

3. Debelle, G. and G. Stevens (1995). Monetary Policy Goals for Inflation in Australia, Reserve Bank of Australia, Research Discussion Paper No. 9530.

4. Jean Claude Trichet, (2004), Federal reserve board's semiannual Monetary Policy Report to the congress round table, July 1, 2004

5. Morling, S. (1997). Modeling Inflation in Fiji, Working Paper, Reserve Bank of Fiji.

APPENDIX 1: DATA DEFINITIONS

This appendix describes the data. The list the definitions of the data and give their sources. All data are quarterly, and the sample period is 1995(1)-2009(2).

2. Consumer Price Index (IPC)

Definition: The Consumer Price Index is a Laspeyres index that covers household consumption as it is used by national accounts. The reference population for the CPI consists of all households living in urban areas in Rwanda. The household basket includes 438 products observed in many places spread all over the administrative centers of all provinces in Rwanda. All kinds of places of observation are selected: shops, markets, services, etc. More than 25,000 prices are collected every month by enumerators of the National Institute of Statistics of Rwanda and of the National Bank of Rwanda. The base year for the CPI is the year 2003. The weights used for this new index are the result of the Household Living Conditions Survey (EICV) conducted in 2000-2001 with a sample of 6,450 households. The splicing with the old index is feasible using the splicing coefficient of 3.889. If you divide the old index by this coefficient, you will be able to make comparisons with the new index based in 2003.

The consumer price index is calculated by the RNIS and published every quarterly the base is 2003=100 (NISR) is adjusted.

Source: National Institute of Statistics of Rwanda (NISR), P.O. BOX 6139 Kigali,

Tel.: (250) 750545 Fax: (250) 575719, Web site: www.statistics.gov.rw /CPI Indexes

3. Unit Labour Costs (ULC)

Definition: Nominal cost of all labour per unit of output.

Nominal unit labour costs are defined as:

salaries + payroll taxes - employment subsidies divided by gross domestic product Where salaries refers to the wages, salaries and supplements of all employee in public and private sector including volontaries wage and salary earners. The class «wage and salary earners» is only a subset of all employed people in the economy,

f-employed, employers. Unit labour costs of wage

scaled up to that by adding the 5% as legal percentage of ratio payed in national social security fund( NSSF).

Source: Author calculation using then NSSF data.

4. Import Prices (IP)

Definition: The NBR publish every month the quantity of goods imported and they values, using these data we calculate the import price index of merchandise imports, excluding without the oil import items. Import prices are measured as the implicit price on seasonally adjusted merchandise imports, excluding exogenous imports. Exogenous imports are goods which are lumpy in nature, subject to government arrangements or significantly affected by factors other than the general level of economic activity in Rwanda. Specifically, this covers fuel, defense equipment, and ships, aircraft and other large items of equipment acquired by selected public and private enterprises.

The monthly data are adjusted on quarterly basis using the moving average, Author calculation

Source: NBR, Foreign Exchange Inspection and Balance of Payments Department

5. Petrol Prices (PP)

Definition: Automotive fuel price index or oil price index.

On the monthly basis the NBR publish the quantity and the values of energy and lubricant which include piles and electrics accumulators, fuels, gaz oils, lubricating oils and others fuels products using these available data we calculate the oil price index on Layperes basis and finally adjusted on quarterly.

Source: NBR, Foreign Exchange Inspection and Balance of Payments Department 5. M2

Definition: M2 is the monetary base which include the currency in circulation plus
commercial bank reserves at the central bank this mean M1( currency in circulation

nd saving deposit which is published by the national s deflated using the current CPI.

Source: NBR, Research department.

6. The exchange rate (TCUSD)

Definition: The nominal exchange rate used in the study is defined as unit of Rwanda franc per United States dollar. The basic calculation is made by dairly basis and later adjusted on the quarterly base.

Source: NBR, Foreign Exchange Inspection and Balance of Payments Department

7. Interest rate (TDC)

Definition: The interest rate represents the spread between the 91-day treasury bills and the average annual interest offered on time and saving deposit and this spread represent the return on domestic financial assets. The interest rate of central has been used as the measure of the spread.

Source: NBR, Foreign Exchange Inspection and Balance of Payments Department

8. The nominal GDP

The GDP represent the national output is published on the year basis we estimated it using the eviews to get the quarterly data. This involves adjusting a linear interpolation of annual GDP series with weigthed percentage errors between actual quarterly data. Then the resulting was seasonally adjusted to minimize the impact of the cyclical components.

PIB

ULC

IP

TCUSD

OIL PRICE INDEX

72.90

0.0064

1082168

172.99

341.84

81.60

0.0063

810447

273.42

347.02

89.00

0.0062

1265720

312.94

339.49

95.50

0.0069

1092288

304.43

439.08

96.10

0.0072

986104

305.16

315.97

102.20

0.0075

3478570

307.14

339.08

109.00

0.0072

663810

306.94

315.97

116.70

0.0072

606585

307.42

315.89

129.70

0.0065

282688

305.35

318.2

137.00

0.0062

329202

301.67

310.61

143.20

0.0063

265033

299.65

319.95

148.30

0.0064

309131

304.13

319.95

150.90

0.0063

317448

307.25

326.39

154.30

0.0063

292589

309.30

328.44

156.90

0.0062

361544

315.44

350.65

159.10

0.0067

359701

325.17

373.87

157.50

0.0069

330049

332.73

403.09

162.20

0.0071

338890

335.24

410.89

165.50

0.0070

319282

334.18

437.56

171.10

0.0076

368840

342.67

458.61

174.80

0.0075

434809

362.87

454.38

178.20

0.0076

377019

379.39

457.77

181.50

0.0076

356082

405.45

459.32

183.10

0.0081

335797

425.22

461.17

186.50

0.0076

394560

433.65

466.04

190.20

0.0076

361482

439.91

466.43

194.30

0.0076

304842

444.86

495.12

199.20

0.0079

329135

456.52

496.18

203.80

0.0081

310087

460.78

517.58

208.40

0.0082

372846

466.90

532.32

213.40

0.0083

395299

482.01

533.5

216.00

0.0087

395426

502.01

563.25

222.20

0.0090

454943

513.80

517.2

229.30

0.0091

430441

531.69

493.6

237.60

0.0090

495160

550.98

500.8

249.70

0.0094

485623

570.24

497.0

259.10

0.0100

455757

582.43

459.5

268.10

0.0099

462887

579.73

388.3

277.20

0.0098

466906

576.23

386.9

285.40

0.0100

593191

568.84

361.7

294.60

0.0098

507757

562.92

420.4

TDC

IPC

PERIOD M2

39.80

10.04

-

49.74

12.06

61.87

59.27

12.04

70.80

62.64

12.12

71.73

62.13

9.00

70.90

64.58

9.36

71.10

66.60

9.43

74.40

68.87

11.26

76.33

74.14

8.39

77.63

84.29

9.07

78.40

81.11

8.52

81.43

88.60

8.24

88.10

85.45

9.94

88.77

82.66

9.21

89.27

84.18

9.16

86.20

91.99

7.76

84.47

102.79

7.91

84.90

103.04

9.19

83.93

107.38

8.53

85.00

102.51

8.75

86.00

103.80

9.20

86.23

109.72

9.44

86.93

107.36

9.41

88.57

119.39

10.11

91.47

122.17

9.69

92.03

127.75

10.16

91.13

125.32

9.91

90.90

130.69

10.18

91.07

129.70

10.15

91.20

135.60

10.00

91.80

135.30

9.34

93.53

144.31

9.10

95.80

143.25

8.94

96.87

145.89

9.11

98.87

158.31

9.27

100.83

167.53

9.33

103.43

162.41

8.91

107.24

161.28

9.26

109.82

167.25

9.33

113.56

187.23

9.43

117.18

190.84

13.50

120.06

1995-1

1996-1

1997-1

1998-1

1999-1

2000-1

2001-1

2002-1

2003-1

2004-1

2005-1

 
 

122.70

 

123.39

 

122.49

220.07

8.08

128.01

237.64

8.18

132.81

254.15

8.32

134.84

285.98

8.24

136.24

273.63

7.90

143.08

301.76

7.69

144.33

326.77

7.57

145.81

375.27

7.32

146.91

422.16

7.56

152.70

445.39

8.21

163.03

465.03

10.10

174.40

468.16

9.23

179.40

468.16

7.90

181.90

468.00

8.10

185.30

303.70

0.0099

549816

556.99

320.1

313.20

0.0095

511935

554.44

230.2

322.20

0.0099

519140

553.73

371.5

332.10

0.0108

486034

553.86

506.3

342.10

0.0109

541193

552.04

364.5

352.80

0.0118

439671

551.29

345.5

358.60

0.0125

452765

550.01

328.4

354.20

0.0135

411030

547.87

399.4

371.85

0.0136

439717

546.37

357.8

387.76

0.0133

556641

547.89

444.5

403.46

0.0137

504527

545.12

605.5

419.54

0.0138

594475

543.89

574.8

517.68

0.0118

755817

543.35

645.4

500.73

0.0126

771322

547.13

574.8

451.19

0.0150

778,849

553.01

621.7

369.12

0.0177

773299

566.00

605.5

515.22

0.0132

721312

572.30

532.0

2006-1

2007-1

2008-1

2009-1

2009-2

APPENDIX 3: RESIDUALS PROPERTIES

1996 1998 2000 2002 2004 2006 2008

.06

.04

.02

.00

-.02

-.04

-.06

.3

.2

.1

.0

-.1

-.2

-.3

-.4

.10
.05
.00

-.05
-.10
-.15

.2

.1

.0

-.1

-.2

DLPIBC Residuals

DLIPC Residuals

1996 1998 2000 2002 2004 2006 2008

DLTDC Residuals

DLM2 Residuals

1996 1998 2000 2002 2004 2006 2008

1996 1998 2000 2002 2004 2006 2008

DLTCUSD Residuals

DLULC Residuals

1996 1998 2000 2002 2004 2006 2008

1996 1998 2000 2002 2004 2006 2008

.20
.15
.10

.05
.00

-.05
-.10
-.15

.03

.02

.01

.00

-.01

-.02

-.03

Page 71

Vector Autoregression Estimates

Date: 11/22/09 Time: 11:49

Sample (adjusted): 1995Q4 2009Q2

Included observations: 55 after adjustments Standard errors in ( ) & t-statistics in [ ]

LIPC

LIPC(-1) 1.101540

(0.26175)

[ 4.20837]

LIPC(-2) -0.422070

(0.27787)

[-1.51896]

LMARKUP(-1) -0.056816

(0.03854)

[-1.47435]

LMARKUP(-2) 0.013979

(0.03739)

[ 0.37387]

LEXCESS_MONEY(

-1) -0.083627
(0.13791) [-0.60639]

LEXCESS_MONEY(

-2) -0.027293
(0.14280) [-0.19113]

LM2(-1) 0.111751

(0.14672)

[ 0.76165]

LM2(-2) 0.031747

(0.16028)

[ 0.19807]

LPIBC(-1) 0.089677

(0.06716)

[ 1.33521]

LEXCESS_

LMARKUP MONEY

LM2

LPIBC

LTCUSD

2.141111

-2.130141

-1.578700

-1.192560

-0.352104

(1.26259)

(0.84661)

(0.65981)

(0.84264)

(0.17829)

[ 1.69581]

[-2.51609]

[-2.39264]

[-1.41526]

[-1.97489]

-3.151490

1.704996

1.269049

-0.239753

0.231334

(1.34034)

(0.89874)

(0.70045)

(0.89453)

(0.18927)

[-2.35127]

[ 1.89709]

[ 1.81177]

[-0.26802]

[ 1.22224]

0.450103

0.170646

0.012606

0.227207

-0.033242

(0.18589)

(0.12464)

(0.09714)

(0.12406)

(0.02625)

[ 2.42140]

[ 1.36908]

[ 0.12977]

[ 1.83144]

[-1.26640]

0.231217

-0.193665

-0.092581

0.010903

-0.035885

(0.18035)

(0.12093)

(0.09425)

(0.12037)

(0.02547)

[ 1.28204]

[-1.60144]

[-0.98230]

[ 0.09058]

[-1.40905]

0.564419 (0.66523) [ 0.84845]

0.032940 (0.44606) [ 0.07385]

-0.761326 (0.34764) [-2.18996]

-0.568270 (0.44397) [-1.27997]

-0.026493 (0.09394) [-0.28202]

-1.341830

0.298516

0.279204

-0.394038

0.055375

(0.68882)

(0.46187)

(0.35997)

(0.45971)

(0.09727)

[-1.94802]

[ 0.64631]

[ 0.77563]

[-0.85714]

[ 0.56931]

-0.810507

0.660783

1.492038

0.765855

-0.076622

(0.70774)

(0.47456)

(0.36986)

(0.47234)

(0.09994)

[-1.14521]

[ 1.39241]

[ 4.03410]

[ 1.62141]

[-0.76668]

1.375501

-0.332027

-0.249299

0.523881

0.027241

(0.77313)

(0.51841)

(0.40403)

(0.51598)

(0.10917)

[ 1.77914]

[-0.64047]

[-0.61704]

[ 1.01531]

[ 0.24952]

-0.013350

0.295970

0.442492

0.448134

0.010639

(0.32397)

(0.21723)

(0.16930)

(0.21622)

(0.04575)

[-0.04121]

[ 1.36245]

[ 2.61359]

[ 2.07261]

[ 0.23257]

LTCUSD(-1) -0.281546

(0.14066)

[-2.00166]

LTCUSD(-2) 0.222899

(0.13707)

[ 1.62619]

C -1.676818

(1.69667)

[-0.98830]

R-squared 0.996371

Adj. R-squared 0.995334

Sum sq. resids 0.013769

S.E. equation 0.018106

F-statistic 960.8409

Log likelihood 150.0073

Akaike AIC -4.982082

Schwarz SC -4.507621

Mean dependent 0.043181

S.D. dependent 0.265052

-0.016498

-0.358248

-0.444169

-0.358673

0.176711

(0.43672)

(0.29283)

(0.22822)

(0.29146)

(0.06167)

[-0.03778]

[-1.22338]

[-1.94619]

[-1.23059]

[ 2.86547]

-0.974949

0.733304

0.431368

0.233600

1.218535

(0.67847)

(0.45494)

(0.35456)

(0.45281)

(0.09581)

[-1.43697]

[ 1.61187]

[ 1.21662]

[ 0.51589]

[ 12.7186]

0.800388

-0.862067

-0.552112

-0.127980

-0.311163

(0.66117)

(0.44334)

(0.34552)

(0.44126)

(0.09336)

[ 1.21056]

[-1.94450]

[-1.59792]

[-0.29003]

[-3.33279]

-0.864729

0.918994

-0.389253

18.86430

-4.492430

(8.18413)

(5.48774)

(4.27694)

(5.46204)

(1.15568)

[-0.10566]

[ 0.16746]

[-0.09101]

[ 3.45371]

[-3.88725]

0.737300

0.627502

0.995287

0.986779

0.998160

0.662242

0.521074

0.993941

0.983002

0.997634

0.320365

0.144041

0.087492

0.142695

0.006388

0.087337

0.058562

0.045641

0.058288

0.012333

9.823162

5.896025

739.2045

261.2321

1898.408

63.46310

85.44554

99.15586

85.70373

171.1258

-1.835022

-2.634383

-3.132941

-2.643772

-5.750031

-1.360561

-2.159923

-2.658480

-2.169312

-5.275570

0.014606

-0.001208

5.025937

28.44066

6.090165

0.150278

0.084622

0.586355

0.447072

0.253544

[-0.47015]

Determinant resid covariance

 

(dof adj.)

4.13E-19

Determinant resid covariance

8.19E-20

Log likelihood

740.3481

Akaike information criterion

-24.08538

Schwarz criterion

-21.23862

Appendix 5: Variance decomposition of VECM

Variance Decomposition of LIPC: Perio

d S.E. LIPC LMARKUP

LEXCESS
_ MONEY

LM2

LPIBC

LTCUSD

1

0.018297

100.0000

0.000000

0.000000

0.000000

0.000000

0.000000

2

0.032540

96.90690

0.261702

0.595329

0.499544

1.721004

0.015519

3

0.043520

90.62316

0.191614

4.832142

2.512064

1.817140

0.023875

4

0.052351

83.63004

0.136518

8.777639

5.243455

1.894989

0.317361

5

0.059778

78.48380

0.195292

12.07487

6.924228

1.525553

0.796257

6

0.066923

75.02531

0.419849

14.20681

7.772816

1.218449

1.356770

7

0.074009

72.88328

0.648207

15.70749

7.907380

1.077777

1.775871

8

0.081031

71.31248

0.750457

16.86669

7.916372

1.034312

2.119692

9

0.087759

70.04459

0.799915

17.85948

7.905447

0.980837

2.409730

74863 18.69154 7.959918 0.875919 2.673639

RKUP:

LEXCESS

MONEY LM2 LPIBC LTCUSD

Perio

d S.E. LIPC LMARKUP

_

1

0.085733

10.12798

89.87202

0.000000

0.000000

0.000000

0.000000

2

0.131525

25.71178

72.02151

0.006302

0.899849

0.264277

1.096287

3

0.163858

29.98207

63.86561

0.275853

0.849396

4.217105

0.809964

4

0.177746

30.92442

60.75896

0.698480

0.730917

5.229595

1.657623

5

0.192141

29.93022

62.02403

0.724407

0.649359

4.550724

2.121260

6

0.214214

28.82513

62.77342

0.613363

0.525383

4.773778

2.488929

7

0.234401

28.50856

63.99687

0.619720

0.452586

3.987878

2.434389

8

0.248487

27.96734

63.66144

0.782854

0.407589

4.605363

2.575415

9

0.259852

27.17995

63.00427

0.842011

0.383605

5.795693

2.794468

10

0.271840

27.12109

63.31215

0.786374

0.399906

5.369609

3.010871

Variance Decomposition of LEXCESS_MONEY:

Perio

d

S.E.

LIPC

LMARKUP

LEXCESS
_ MONEY

LM2

LPIBC

LTCUSD

1

0.066526

37.38368

0.041603

62.57472

0.000000

0.000000

0.000000

2

0.095655

40.93250

3.918057

53.35775

1.232881

0.537617

0.021192

3

0.117338

44.87656

3.191635

50.20760

0.825644

0.747469

0.151092

4

0.135264

47.85993

2.807033

46.95173

0.631856

1.615467

0.133980

5

0.149603

50.12986

2.460321

45.22608

0.531474

1.481649

0.170621

6

0.161253

50.98821

2.647058

44.45449

0.484415

1.275346

0.150483

7

0.171989

51.31621

2.706138

44.27728

0.428598

1.134399

0.137376

8

0.183281

51.89167

2.584419

43.92870

0.377480

1.096480

0.121256

9

0.194634

52.81078

2.376649

43.25861

0.336331

1.102984

0.114654

10

0.204847

53.63574

2.288540

42.62055

0.306779

1.039664

0.108725

Variance Decomposition of LM2: Perio

d S.E. LIPC LMARKUP

LEXCESS
_ MONEY

LM2

LPIBC

LTCUSD

1

0.051043

0.350690

0.055566

89.42647

10.16727

0.000000

0.000000

2

0.073794

0.264612

1.755991

76.19698

19.86784

1.871757

0.042818

3

0.094336

0.678710

1.849955

75.90141

20.13481

1.184020

0.251098

4

0.111029

0.990596

2.250991

73.81612

21.58103

0.911013

0.450254

5

0.125225

1.091928

2.660878

73.37158

21.37689

0.776895

0.721820

6

0.137984

0.961006

3.373420

72.87852

21.27225

0.652282

0.862522

7

0.150007

0.835199

3.709558

72.80418

21.07802

0.593375

0.979670

8

0.161627

0.761029

3.813367

72.71875

21.01232

0.633132

1.061403

9

0.172552

0.731259

3.834466

72.64545

20.98426

0.636342

1.168230

10

0.182762

0.693408

3.985485

72.54635

20.92388

0.593409

1.257465

Variance Decomposition of LPIBC:

Perio

d S.E. LIPC LMARKUP

LEXCESS

MONEY LM2 LPIBC LTCUSD

_

22762
29179
04580

4 0.098952 14.50754 37.38808

5 0.102642 15.49777 38.18783

6 0.116172 18.32080 38.82996

7 0.130118 20.95032 39.69009

8 0.136354 21.16591 39.71859

9 0.140460 20.28022 39.00492

10 0.143889 20.17289 40.02714

0.000290

7.565386

59.66085

0.000000

0.689948

5.504930

39.01509

0.361021

2.706326

7.924112

31.11456

0.446975

4.071785

8.757723

34.88760

0.387276

4.196817

9.282737

32.44972

0.385130

3.372594

7.782297

31.39360

0.300751

3.178140

7.329111

28.57389

0.278452

4.142156

8.617926

26.08331

0.272105

4.870003

9.756058

25.82769

0.261103

4.902321

10.00941

24.61460

0.273639

Variance Decomposition of

LTCUSD:

Period LEXCESS

S.E. LIPC LMARKUP MONEY LM2 LPIBC LTCUSD

_

1

0.011168

0.173076

7.267890

8.255018

1.279036

4.236467

78.78851

2

0.022223

1.053871

11.07978

18.00470

1.116873

7.527846

61.21693

3

0.035166

3.220732

13.36857

23.19984

0.463252

12.33115

47.41645

4

0.050203

4.226496

10.96903

26.08153

0.228007

20.42038

38.07456

5

0.065036

4.829875

9.019364

27.59094

0.143855

24.71258

33.70338

6

0.078331

5.752720

8.409146

29.00032

0.109422

24.39696

32.33143

7

0.090758

6.952437

8.811006

30.28924

0.113849

22.24306

31.59041

8

0.103117

7.713250

9.178553

31.46720

0.170577

20.71041

30.76000

9

0.115532

7.744092

9.075099

32.37502

0.258851

20.64557

29.90137

10

0.127537

7.499270

8.803140

32.89073

0.319129

21.15154

29.33619

Cholesky Ordering: LIPC LMARKUP LEXCESS_MONEY LM2
LPIBC LTCUSD

 

its test

1.5 1.0 0.5 0.0 -0.5

-1.0

00 01 02 03 04 05 06 07 08

Recursive C(1) Estimates #177; 2 S.E.

.05 .00 -.05 -.10 -.15 -.20 -.25

-.30

00 01 02 03 04 05 06 07 08

.2 .0 -.2 -.4

-.6

00 01 02 03 04 05 06 07 08

Recursive C(2) Estimates #177; 2 S.E.

.6 .4 .2 .0

-.2

00 01 02 03 04 05 06 07 08

.25 .20 .15 .10 .05 .00 -.05

-.10

00 01 02 03 04 05 06 07 08

Recursive C(3) Estimates #177; 2 S.E.

.2
.1
.0

-.1

-.2

-.3

00 01 02 03 04 05 06 07 08

0.4 0.0 -0.4 -0.8

-1.2

00 01 02 03 04 05 06 07 08

Recursive C(4) Estimates #177; 2 S.E.

.6 .4 .2 .0 -.2

-.4

00 01 02 03 04 05 06 07 08

Recursive C(5) Estimates #177; 2 S.E.

Recursive C(6) Estimates #177; 2 S.E.

Recursive C(7) Estimates #177; 2 S.E.

Recursive C(8) Estimates #177; 2 S.E.

.8 .4 .0 -.4

-.8

Recursive C(9) Estimates #177; 2 S.E.

00 01 02 03 04 05 06 07 08

.3 .2 .1 .0

-.1

-.2

Recursive C(10) Estimates #177; 2 S.E.

00 01 02 03 04 05 06 07 08

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