Int. J. Monetary Economics and Finance, Vol. 4, No. 4, 2011

355

Impact of exchange rate and money supply on growth, inflation and interest rates in the UK Keshab Bhattarai Business School, University of Hull, HU6 7RX, HULL, UK Fax: 01482-463484 E-mail: [email protected] Website: http://www.hull.ac.uk/php/ecskrb. Abstract: Growth rates, inflation and interest rates are determined simultaneously in the UK. Depreciations of Sterling pounds contribute to the growth by enhancing international competitiveness. Inflation from the growth of money, depreciation of Sterling and higher interest rates, impacts adversely on it. London being a hub of the global financial market higher interest rates are persistent and coexist with greater liquidity of the financial system, making money supply non-neutral in the short run as in Desai and Weber (1988), Fisher and Whitley (2000), Mellis and Whittaker (2000), Wallis (1969, 1989) for the UK and Sargent (1976) and Fair (1993). Keywords: simultaneous equation model; macromodel. Reference to this paper should be made as follows: Bhattarai, K. (2011) ‘Impact of exchange rate and money supply on growth, inflation and interest rates in the UK’, Int. J. Monetary Economics and Finance, Vol. 4, No. 4, pp.355–371. Biographical notes: Keshab Bhattarai (PhD Northeastern) originally from Jamune 7, Tanahun, Nepal, has been lecturing and supervising students on micro, macro, econometrics, economic analysis, empirical economics and research methods for PhD/MSc/BSc/BA programs for more than a decade at the Business School of University of Hull in UK. He has published articles in major international journals and written several books. He taught economics at Northeastern before receiving the PhD in 1997 and before the post-doctoral research at the Warwick University. He holds MA from the ISS, Hague, Netherlands and the TU in Nepal and had worked for the Government of Nepal and a commercial Bank in 1980s.

1

Introduction

Variations in economic activities of households and firms in UK have caused significant fluctuations in growth and employment. Unemployment rates have risen after each recession since early 1970s as shown in Figure 1. These fluctuations that reflect changes in the demand and supply sides owing to continuous shifts in confidence of consumers and producers in the economy have caused changes not only in the price level but also in the interest and exchange rates (Figures 2 and 4). These also have frequently resulted in Copyright © 2011 Inderscience Enterprises Ltd.

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stagflationary situations (Figures 3 and 5) with occurrence of high inflation and high unemployment rate simultaneously. While growth rates of money and output seem to be moving together (Figure 6) whether the depreciation of exchange rates, on average 0.4% per quarter since 1970, is systematically related with the growth and hence unemployment rate is an interesting empirical issue. Figure 1

Unemployment and growth rates in UK (see online version for colours)

Figure 2

Inflation and interest rates in UK (see online version for colours)

Figure 3

Growth rate and inflation in UK (see online version for colours)

Impact of exchange rate and money supply Figure 4

Exchange of sterling pound to the US Dollar (see online version for colours)

Figure 5

Inflation and unemployment rate in UK (see online version for colours)

Figure 6

Growth rates of output and money in UK (see online version for colours)

357

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K. Bhattarai

On the theoretical side, there is a considerable controversy on whether the growth rates of output are influenced by demand-side factors in the short run as wage contracts and asymmetric information prolong the adjustment towards the long-run equilibrium as studied by Driffill and Schultz (1992), despite more unanimity among economists about the role of supply-side factors such as the physical and human capital and the technology on growth in the long run. Keynesian and new Keynesian models show positive impacts of money in the growth and employment rates in contrast to neutrality of money in the classical and new classical models (Desai and Weber, 1988). For policy-makers, empirical evidence contained in time series as given in Figures 1–6 are testing grounds for theories while setting fiscal, monetary or exchange rate policies to stabilise output and employment using the interest rate, money supply and exchange rates as instruments to achieve those objectives (Fisher and Whitley, 2000; Mellis and Whittaker, 2000). Establishing a precise relation between the stabilisation objectives and fiscal and monetary policy instruments is the main contribution of a macroeconometric simultaneous equation modelling in the literature for this purpose (Wallis, 1969, 1989; Sargent, 1976; Taylor, 1987; Fair, 1993; Holly and Weale, 2000; Garratt et al., 2003). Most correlations among these macroeconomic series presented in Table 1 are theoretically plausible. While the positive Correlation Of Inflation (CPI) with the Exchange Rate (ESRT$), treasury bill rate (Treasury) and growth rate of money supply (gr_M2) is theoretically justifiable, the negative correlation of CPI with the growth rate (gr_GDP) and a positive correlation with unemployment (unempl2) cast doubt on the aggregate supply or Phillips curve type reasoning. Similarly, positive correlations of the treasury rate with the growth rate of money supply and GDP is according to the theoretical predictions but its positive correlation with the unemployment rate is as not as one would expect on theoretical basis. In the same way, positive correlations of growth rate with the exchange rate, interest rate and growth rate of money supply are theoretically plausible but its negative correlation with inflation only supports the stagflationary hypothesis. These correlations just indicate that cause–effect relations in the economy are much complicated than suggested by these simple correlations. Table 1

Correlation among macro time series CPI

EXRT$

Treasury

gr–M2

gr_gdp

CPI

1

EXRT$

0.246636

1

Treasury

0.705762

0.237376

1

gr–M2

0.128238

0.069048

0.232785

1

gr_gdp

–0.36207

0.021627

0.153351

0.166649

1

unempl2

0.43822

–0.22536

0.268436

–0.01415

–0.07706

unempl2

1

As growth rates, inflation, interest rate and exchange rates are simultaneously related, a three-equation simultaneous equation model outlined in the next section is an appropriate model to study such simultaneity issue. Further details on analytical solutions for 12 reduced form coefficients along with the rank and order conditions to retrieve those structural coefficients are given in Appendix 2 followed by the derivation and asymptotic properties of the ILS, two Stage Least Square (2SLS) and three Stage Least Squares (3SLS) estimators based on the literature in econometric theory in Appendix 3.

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Analysis of results from the estimations of ILS, 2SLS and 3SLS is analysed briefly in Section 3 followed by conclusions and references and tables of results at the end.

2

A simultaneous equation model of growth, inflation and interest rates

The objective of the model is to analyse interdependence among quarterly growth rates (gt), interest rates (Rt) and inflation (πt) in the UK taking growth rate of money supply (Mt), exchange rate (Et) and lagged interest rate (Rt–1) among its determinants. The first equation relates growth rate of output to the interest rate in the current and previous periods and the current exchange rate. This can be interpreted as an IS curve for an open economy. Interest rates reflect the cost of capital; the higher the cost of capital the lower will be the growth rate. A higher growth rate of money is expansionary and should raise output. Similarly, a higher exchange rate makes economy less competitive in the global economy and thus has negative impact on output. This is exactly what is seen in the empirical results. Second equation links current real interest rate to the previous interest rate, exchange rate and money supply. This can be considered a version of interest rate rule. In a time of greater liquidity, the interest rate is expected to fall with a higher rate of growth of money supply normally but this may be positive if London remains a hub of the global financial system as seen in the empirical finding. The lagged interest term helps to determine the persistent pattern in the interest rates indicating time lags between the decision and impact of changes in the interest rate. The third equation relates inflation to domestic supply factors represented by growth rates and foreign demand-side factors captured in the exchange rates and the growth rate of money supply. As expected, inflation is higher with higher growth rates of output and money supply but lower with depreciation. Equations for this three-equation simultaneous equations model could be presented as follows: Output growth equation: gt = α 0 + α 2 Rt + α 3 Et + α 5 Rt −1 + e1,t

(1)

Interest rate equation: Rt = γ 0 + γ 2π t + γ 3 M t + γ 5 Rt −1 + e3,t

(2)

Inflation rate equation: π t = β 0 + β1 gt + β3 Et + β 4 M t + e2,t .

(3)

Here, gt, πt and Rt are three endogenous variables of the system and Et, Mt and Rt–1 are three exogenous or pre-determined variables. The error terms, e1,t, e2,t and e3,t, are identically and normally distributed noises with zero mean and constant variance. Coefficients α0, α2, α3, α5, β0, β1, β3, β4, γ0, γ2, γ3 and γ5 are 12 structural parameters that remain to be estimated. As the application of the OLS technique to each individual equation results in biased and inconsistent estimates (proof in section Appendix 3), the system of equations needs to be estimated simultaneously. This involves four distinct steps: •

forming a reduced form of the system

•

estimating the reduced form coefficients

•

retrieving the structural coefficients of the model

•

using those parameters in prediction and forecasts.

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Before that, it is necessary to ensure that each variable used in regression is stationary based on ADF tests. For this reason, model reported here used stationary variables; the growth rate of GDP (DLRGDP), growth rate of money supply (DLM4) and inflation (CPI) were stationary and the percentage changes in the treasury bills rate (DLTbills) and exchange rates (DLExrt_dllr) (see ADF test statistics in Appendix 3). Then, i each equation of the model is identified using the rank and order conditions, so that it is possible to retrieve the structural coefficients from the estimates of the reduced form coefficients. For a rank test, a matrix of structural coefficients formed is shown in Table 2: Table 2

Matrix of coefficients of the system for rank condition Constant

gt

πt

Rt

Et

Mt

Rt–1

Growth

–α0

1

0

–α 2

–α3

0

–α5

Interest rate

–γ5

0

–γ 2

1

0

–γ 4

–γ5

Inflation

–β0

–β1

1

0

–β3

–β4

0

Thus, each of the above-mentioned equation is identified by the rank condition as:  −γ For growth equation: A1 =  2  1

−γ 4  −γ A1 = 2  β4  1

1  1 −α 3  For interest rate: A2 =   A2 = 1 β β − β 1  1 3 

−α 3

β3

−γ 4

β4

= − β 4γ 2 + γ 4 ≠ 0

= − β 3 + β1α 3 ≠ 0

−α 2 −α 5  −α −α  For inflation rate: A3 =  2 5  A3 = = α 2γ 5 + α 5 ≠ 0. 1 −γ 5  1 −γ 5 

This means the structural coefficients can be retrieved from the estimates of the reduced form coefficients. Order condition involves checking the identifiability condition K − k ≥ m − 1 where K is the number of exogenous variable in the system including the intercept term, k is the number of exogenous variables in the equation under consideration and m is the number of endogenous variables in that equation. The number of exogenous variables in this system is six, K = 6, i.e.,α0, β0, γ0, Et, Mt and Rt–1. Order condition is satisfied for each of the above-mentioned equations K − k = 6 − 3 = 3 ≥ m − 1 = 3 − 1 = 2 . Therefore, each of the above-mentioned equations is overidentified. This means it is possible to retrieve more than one value of the structural coefficients from the estimates of the reduced form coefficients. The procedure is shown in Appendices 1 and 2.

3

Analysis of results from the ILS, 2SLS and 3SLS estimations

Parameters of the model, estimated using ILS and 2SLS, are presented in Table 3 and those estimated using 3SLS are in Table 4. Basic hypotheses regarding the determinants

Impact of exchange rate and money supply

361

of growth rate, inflation of the retail price index and the interest rate are supported by these estimates. There is a negative relation between growth rates and the exchange rates (DLExrt_dllr). Lower exchange rates are associated with higher growth rates. Continued depreciation of the Sterling Pounds from 2.5 in 1972 to 1.11 in 1984 and then a bit appreciation towards 1.5 in 2000 and 1.8 in 2008 has been one important factor influencing the rate of economic growth. Depreciation of pound in this manner has made the UK economy more competitive in the world and this has contributed to a higher growth rate. Higher level of liquidity has promoted growth as reflected in positive contribution of growth rate of money supply (DLM4) in the growth rate of output. One percent change in the growth rate of money brings 0.6% increase in the growth rate of output. These estimates do not support the neutrality of money hypothesis. Higher interest rate in the previous period (DLTbill_1), which can be considered as an instrument of contractionary economic policy, has negative impact, but the coefficient is not significant. Model estimates suggest persistency in the quarterly interest rates. About 15% of current interest rate is explained by previous interest rate. This is consistent to finding that the UK economy seems to be following implicitly an interest rate rule since early 1970s (Bhattarai, 2008). It is a bit surprising to note a positive relation between the interest rate and growth rate of money. The higher the liquidity in the system, the higher is the interest rate. Given the prominent place of London as a financial centre of the world, this might just indicate that speculative demand dominates the transaction demand for money in the UK’s financial system. Similarly, the association between the exchange rate and the interest rate is negative and significant. Depreciation of pound or appreciation of foreign currency results in higher rate of interest rate in the UK. The outflow of capital is more likely to occur if Pound continues to depreciate. There seem to be a significant relation between the growth rate of money supply and inflation. Higher rate of growth of money leads to a higher inflation. This result is sensible in the light of the quantity theory of money. Depreciation of pound has positive impact on inflation. This is also sensible as the depreciation makes imports more expensive and prices of imported commodity to rise leading to a further increase in inflation. Similarly, higher interest rates have positive impact on prices. A higher interest rate raises the cost of capital and hence the inflation. All three estimation techniques suggest that the money supply has remained a significant determinant of growth rate, inflation and the interest rate during the study period. Table 3

Estimations of reduced form coefficients by ILS and 2SLS methods Indirect least square growth equation Coefficient

Std.Error

t-value

t-prob

DLTbills_1

–0.0032

0.0258

–0.1230

0.9020

DLExrt_dllr

–0.0187

0.0725

–0.2580

0.7970

DLM4

0.6349

0.1940

3.2700

0.0010

Constant

0.0047

0.0062

0.7610

0.4480

362 Table 3

K. Bhattarai Estimations of reduced form coefficients by ILS and 2SLS methods (continued) Interest rate equation

DLTbills_1 DLExrt_dllr DLM4 Constant

Coefficient

Std.error

t-value

t-prob

0.1503 –0.6302 1.9670 –0.0568

0.0752 0.2114 0.5655 0.0179

2.0000 –2.9800 3.4800 –3.1700

0.0470 0.0030 0.0010 0.0020

Inflation equation DLTbills_1 DLExrt_dllr DLM4 Constant AIC HQ

Coefficient

Std.error

t-value

t-prob

0.6128 –2.7670 30.6555 0.8351

0.8585 2.4130 6.4530 0.2047

0.7140 –1.1500 4.7500 4.0800

0.4760 0.2530 0.0000 0.0000

–10.1402 –10.0461

SC FPE

–9.9086 0.0000

Two-stage least square estimates Growth equation DLTbills_1 DLExrt_dllr DLM4 Constant

Coefficient

Std.error

t-value

t-prob

–0.0032 –0.0187 0.6349 0.0047

0.0258 0.0725 0.1940 0.0062

–0.1230 –0.2580 3.2700 0.7610

0.9020 0.7970 0.0010 0.4480

Interest rate equation DLTbills_1 DLExrt_dllr DLM4 Constant

Coefficient

Std.error

t-value

t-prob

0.1503 –0.6302 1.9670 –0.0568

0.0752 0.2114 0.5655 0.0179

2.0000 –2.9800 3.4800 –3.1700

0.0470 0.0030 0.0010 0.0020

Inflation equation DLTbills_1 DLExrt_dllr DLM4 Constant

Coefficient

Std.error

t-value

t-prob

0.6128 –2.7670 30.6555 0.8351

0.8585 2.4130 6.4530 0.2047

0.7140 –1.1500 4.7500 4.0800

0.4760 0.2530 0.0000 0.0000

AIC

–10.2569

SC

–10.0253

HQ

–10.1629

FPE

0.0000

Impact of exchange rate and money supply Table 4

363

Estimations of reduced form coefficients 3SLS method Growth equation Coefficient

Std.error

t-value

t-prob 0.9020

DLTbills_1

–0.0032

0.0258

–0.1230

DLExrt_dllr

–0.0187

0.0725

–0.2580

0.7970

DLM4

0.6349

0.1940

3.2700

0.0010

Constant

0.0047

0.0062

0.7610

0.4480

Interest rate equation Coefficient

Std.error

t-value

t-prob

DLTbills_1

0.1503

0.0752

2.0000

0.0470

DLExrt_dllr

–0.6302

0.2114

–2.9800

0.0030

DLM4

1.9670

0.5655

3.4800

0.0010

Constant

–0.0568

0.0179

–3.1700

0.0020

t-value

t-prob

Inflation equation Coefficient

Std.error

DLTbills_1

0.6128

0.8585

0.7140

0.4760

DLExrt_dllr

–2.7670

2.4130

–1.1500

0.2530

DLM4

30.6555

6.4530

4.7500

0.0000

Constant

0.8351

0.2047

4.0800

0.0000

AIC

–10.0094

SC

–9.8357

HQ

–9.9389

FPE

0.0000

The above-mentioned estimates are from the quarterly data 1970:2 to 2006:1 obtained from the Office of the National Statistics (ONS) and each variable was properly checked for stationarity (Dickey and Fuller, 1979; Johansen, 1988). Notations DLTbills_1, DLExrt_dllr and DLM4 represent log difference of treasury bills, log difference of exchange rate of pound to US dollars and log difference of money supply each denoting the percentage change in those variables. Variables used in the estimation were stationary or cointegrated and were PcGive econometric software (Doornik and Hendry, 2003) was used for all estimations. Comparison of prediction errors and AIC criteria suggests that 2SLS and 3SLS estimates are better than ILS estimator though the estimates of coefficients in all three methods are quite close to each other.

4

Conclusion

Depreciation of Sterling has contributed to growth of output in UK by enhancing the international competitiveness and so has the higher rate of growth of money supply. Interest rates have been persistent and contractionary and money has been non-neutral in

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the short run. According to these estimates, the higher the liquidity in the financial system the higher is the interest rate. This indicates the role of London as a financial hub in the global economy. Inflation, driven up by growth rate of money as well as the depreciation of pounds and higher interest rates, has adversely affected the growth rate of output. Despite empirical evidence for the persistency in the interest rate, the depreciation of Sterling pounds (or the appreciation of foreign currency) has contributed positively to economic growth. High degree of non-linearity of the reduced form coefficients in terms of structural coefficients in a three-equation simultaneous equation model are expressed in terms of analytical solutions. These were estimated using ILS, 2SLS and 3SLS techniques to show simultaneity among growth rates, inflation, interest rate and exchange rates in the UK. Fresh empirical evidence discussed here are similar to those found in Desai and Weber (1988), Fisher and Whitley (2000), Mellis and Whittaker (2000), Wallis (1969, 1989) for the UK and Sargent (1976) and Fair (1993) for the US economy.

Acknowledgements I appreciate the comments from an anonymous referee of this journal, participants at 41st MMF Annual Conference in Bradford in 2009 and the guest editor of this issue, Sushanta Mallick, for their comments and suggestions, but I take sole responsibility for any errors or omissions that might yet remain in the paper.

References Balke, N.S. and Slottje, D.J. (1993) ‘Poverty and change in the macroeconomy: a dynamic macroeconometric model’, The Review of Economics and Statistics, Vol. 75, No. 1, February, pp.117–122. Bhattarai, K. (2011) Econometric Analysis, Hull University Business School, Workbook. Bhattarai, K. (2008) ‘An empirical study of interest determination rules’, Applied Financial Economics, Vol. 18, No. 4, March, pp.327–343. Davidson, J. (2000) Econometric Theory, Basil Blackwell Publishers, Oxford, UK. Desai, M. and Weber, G. (1988) ‘A Keynesian macro-econometric model of the UK: 1955–1984’, Journal of Applied Econometrics, Vol. 3, No. 1, January, pp.1–33. Dickey, D.A. and Fuller, W.A. (1979) ‘Distribution of the estimator for autoregressive time series with a unit root’, Journal of the American Statistical Association, Vol. 74, pp.427–431. Doornik, J.A. and Hendry, D.F. (2003) PC-Give Volume I-III, Timberlake Consultants Limited, London. Driffill, J. and Schultz, C. (1992) Wage Setting and Stabilization Policy in a Game with Renegotiation, Oxford Economic Papers, New Series, Vol. 44, No. 3 July, pp.440–459. Fair, R.C. (1993) ‘Testing macroeconometric models’, The American Economic Review, Papers and Proceedings, Vol. 83, No. 2, May, pp.287–293. Fisher, P. and Whitley, J. (2000) ‘Macroeconomic models at the Bank of England’, Holly, S. and Weale, M. (Eds.) Econometric Modelling: Techniques and Applications, Chapter 7, The Cambridge University Press, Oxford, UK. Garratt, A., Lee, K., Pesaran, M.H. and Shin, Y. (2003) ‘Long run structural macroeconometric model of the UK’, The Economic Journal, Vol. 113, No. 487, April, pp.412–455. Hendry, D.F. (1997) Dynamic Econometrics, Oxford University Press, Oxford, UK. Holly, S. and Weale, M. (Eds.) (2000) Econometric Modelling: Techniques and Applications, the Cambridge University Press, Cambridge, UK.

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Johansen, S. (1988) ‘Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models’, Econometrica, Vol. 59, No. 6, pp.1551–1580. Judge, G.G., Griffiths, W.E., Hill, R.C., Lutkepohl, H. and Lee, T.C. (1985) Theory and Practice of Econometrics, John and Wiley and Sons, New York. Mellis, C. and Whittaker, R. (2000) Treasury’s forecast of GDP and RPI: how have they changes and what are the uncertainties’, Chapter 3 in Holly, S. and Weale, M. (Eds.): Econometric Modelling: Techniques and Applications, the Cambridge University Press, Cambridge, UK. Sargent, T.J. (1976) ‘A classical macroeconometric model for the United States’, The Journal of Political Economy, Vol. 84, No. 2, April, pp.207–238. Taylor, M.P. (1987) ‘On the long run solution to dynamic econometric equations under rational expectation’, Economic Journal, Vol. 97, No. 385, pp.215–218. Wallis, K.F. (1969) ‘Some recent developments in applied econometrics: dynamic models and simultaneous equation systems’, Journal of Economic Literature, Vol. 7, No. 3, September, pp.771–796. Wallis, K.F. (1989) ‘Macroeconomic forecasting: a survey’, Economic Journal, Vol. 99, March, pp.28–61.

Appendix 1 Retrieving structural coefficients from the reduced form coefficients On methodological side, the superiority of estimators of a simultaneous equation model over single equation models is well known in the literature (Judge et al., 1985; Hendry, 1997). The explicit solutions of structural parameters from the reduced form coefficients and proofs of the above propositions are either limited to two equations model or are presented abstractly in matrix notations (Davidson, 2000). Applied econometricians, however, often have more than two equations in their models and need explicit solutions in order to explain the impacts of one variable on other variables. Since the reduced form coefficients are highly non-linear in the structural parameters, the explicit solution becomes more complex as more equations are added to a model and more so when estimated parameters have more than one solutions. This point is illustrated here using a structural macro-econometric model which involves endogenous relation among the growth rates, inflation and the interest rate with the exchange rate, growth rate of money and the lagged interest rates in the set of exogenous or policy variables. Rearrange the above equations for endogenous and exogenous variables as following: gt − α 2 Rt + 0π t = α 0 + α 3 Et + α 5 Rt −1 + e1,t

(4)

− β1 gt + 0 Rt + π t = β 0 + β 3 Et + β 4 M t + e2,t .

(5)

−0 gt + Rt − γ 2π t = γ 0 + γ 4 M t + γ 5 Rt −1 + e3,t

(6)

 1 −β  1  0

−α 2 0 1

0   gt  α 0 α 3 1   Rt  =  β 0 β 3 −γ 2  π t   γ 0 0

0

β4 γ4

 1    e1,t  E t   + e  0   M t   2,t  γ 5     e3,t   Rt −1   

α5  

(7)

366

K. Bhattarai  gt   1  R  = −β  t  1 π t   0

−α 2

0  1  −γ 2 

0 1

−1

α 0 α 3 β β 3  0  γ 0 0

0

β4 γ4

 1   1 Et     + −β 0  Mt   1  γ5     0  Rt −1 

α5  

−α 2 0 1

0  1  −γ 2 

−1

 e1,t    e2,t   e3,t 

Y = A–1 BX + A–1e     gt   1 R  = − t   (α β γ − 1)  2 1 2 π t          1 − + (α 2 β1γ 2 − 1)     

−β − 1 0

0 1 1 −γ 2 −α 2

0

1

−γ 2

−α 2 0

0 1

0 1 1 −γ 2 −α 2 1

1

0

0

1

 gt   −1 1 R  =   t  (α β γ − 1)  − β1γ 2 2 1 2 π t   − β1

0

0 −γ 2 1 0 − β1 1

−

− β1 0

−

0 −γ 2

−α 2

1 −γ 2

1 −γ 2

1 0 0 −γ 2 −

1 0 − β1 1

−α 2γ 2 −γ 2

 −1 −β γ +  (α 2 β1γ 2 − 1)  − β1 2 1  1

1 −α 2γ 2 −γ 2 1

− β1 0   0 1  1 −α 2   − 0 1   1 −α 2  0  − β1

'

− β1 0   0 1  1 −α 2   − 0 1   1 −α 2  0  − β1

−α 2  α 0 α 3 −1   β 0 β 3 β1α 2   γ 0 0

α 0 α 3 β β 3  0  γ 0 0

0

β4 γ4

 1  E  0   t  M  γ 5   t   Rt −1 

α5  

(8) '

 e1,t    e2,t   e3,t   

0

β4 γ4

 1  E  0   t  M  γ 5   t   Rt −1 

α5  

−α 2   e1,t    −1   e2,t  . β1α 2   e3,t 

(9)

This is the reduced form of the above model. More explicitly it can be written as:  gt   −α 0 − α 2 β 0γ 0 − α 2γ 0 1 R  =   t  (α β γ − 1)  −α 0 β1γ 2 − γ 2 β 0 − β1γ 0 2 1 2 π t   α 0 β1 + β 0 + γ 0 β1α 2

+

−α 3 − α 2 β 3γ 2

−α 2γ 2 β 4 − α 2γ 4

−α 3 β1γ 2 − γ 2 β3 α 3 β1 − β3

−γ 2 β 4 − γ 4 β 4 + γ 4 β1α 2

 1   E  −γ 2 β1α 5 − γ 5   t  M  − β1α 5 + γ 5 β1α 2   t   Rt −1  −α 5 − α 2γ 5

 −e1,t − α 2γ 2 e2,t − α 2 e3,t    −β γ e − γ e − e (α 2 β1γ 2 − 1)  − β 1e 2 +1,t e +2 β2,t α e3,t  1 2 3,t   1 1,t 2,t 1

(10) gt =

−α 0 − α 2 β 0γ 0 − α 2γ 01 −α 3 − α 2 β3γ 2 −α 5 − α 2γ 5 −α γ β − α 2γ 4 + E + 2 2 4 M + R +v (α 2 β1γ 2 − 1) (α 2 β1γ 2 − 1) t (α 2 β1γ 2 − 1) t (α 2 β1γ 2 − 1) t −1 1t

(11)

Impact of exchange rate and money supply

367

Rt =

−α 0 β1γ 2 − γ 2 β 0 − β1γ 0 −α 3 β1γ 2 − γ 2 β3 −γ β α − γ γ β −γ E + 2 4 4 M + 2 1 5 5 R + v (12) + (α 2 β1γ 2 − 1) (α 2 β1γ 2 − 1) t (α 2 β1γ 2 − 1) t (α 2 β1γ 2 − 1) t −1 2t

π1 =

α 0 β1 + β 0 + γ 0 β1α 2 −α 3 β1 − β3 −β α + γ β α β + γ 4 β1α 2 + Et + 4 M t + 1 5 5 1 2 Rt −1 + v3t α β γ − α β γ − α β γ − 1 1 1 ( 2 12 ) ( 2 12 ) ( 2 12 ) (α 2 β1γ 2 − 1)

(13)

where v1t =

−e1,t + α 2γ 2 e2,t − α 2 e3,t

(α 2 β1γ 2 − 1)

; v2t =

− β1γ 2 e1,t − γ 2 e2,t − e3,t

(α 2 β1γ 2 − 1)

; v3t =

− β1e1,t + e2,t + β1α 2 e3,t

(α 2 β1γ 2 − 1)

.

This is written more compactly in the reduced form system as: gt = Π10 + Π11 Et + Π12 M t + Π13 Rt −1 + v1t

(14)

Rt = Π 20 + Π 21 Et + Π 22 M t + Π 23 Rt −1 + v2t

(15)

π t = Π 30 + Π 31 Et + Π 32 M t + Π 33 Rt −1 + v3t

(16)

where Π10 =

−α 0 − α 2 β 0γ 0 − α 2γ 01 −α 3 − α 2 β 3γ 2 −α 2γ 2 β 4 − α 2γ 4 ; Π11 = ; Π12 = ; (α 2 β1γ 2 − 1) (α 2 β1γ 2 − 1) (α 2 β1γ 2 − 1)

Π13 =

−α 5 − α 2γ 5 −α 0 β1γ 2 − γ 2 β 0 − β1γ 0 −α 3 β1γ 2 − γ 2 β3 ; Π 20 = ; Π 21 = ; (α 2 β1γ 2 − 1) (α 2 β1γ 2 − 1) (α 2 β1γ 2 − 1)

Π 22 =

−γ 2 β 4 − γ 4 −γ β α − γ α β + β 0 + γ 0 β1α 2 ; Π 23 = 2 1 5 4 ; Π 30 = 0 1 ; α β γ 1 α β γ 1 − − ( 2 12 ) ( 2 12 ) (α 2 β1γ 2 − 1)

Π 31 =

−α 3 β1 − β3 −β α + γ β α β + γ 4 β1α 2 ; Π 32 = 4 ; Π 33 = 1 5 5 1 2 ; (α 2 β1γ 2 − 1) (α 2 β1γ 2 − 1) (α 2 β1γ 2 − 1)

(17)

Thus it is possible to retrieve 12 structural coefficients from the 12 reduced form coefficients though they are highly non-linear in the structural parameters. Solving larger model requires Gauss Seidel algorithm Fair (1993).

Appendix 2 Efficiency of 3SLS over other estimation techniques Parameters of a simultaneous equation model can be estimated by a number of single or multiple equation estimation techniques. Indirect Least Square (ILS), two Stage Least Square (2SLS), three Stage Least Square (3SLS) and Full Information Maximum Likelihood (FIML) estimators are single equation techniques are prominent ones in the literature (Judge et al., 1985). Indirect least square method involves applying the least square method to the reduced form equations and retrieving the structural coefficients using equation (17). It the matrix algebra the estimates of the above reduced form equation gt can be written as ˆ = ( X ′X )−1 ( X ′Y ) Π

(18)

368

K. Bhattarai

where  N   Et ∑ t ( X ′X ) =   ∑ Mt  t  ∑ Rt −1  t

∑E ∑M ∑E ∑E M ∑E M ∑M ∑E R ∑M R t

t

t

t

2 t

t

t

t

2 t

t

t

t

t

t

t

t

t

t −1

t

t

∑R ∑E R ∑M R ∑R

t −1

t

t

t

   t t −1  ; t t −1   2  t −1  t −1

 ∑ Yt   t   YE  ∑ t t  ( X ′Y ) =  t .  ∑ Yt M t   t   ∑ Yt Rt −1   t 

(19)

It is obvious that these reduced form coefficients can be used to estimate the structural coefficients of the growth equation if the determinant of (X′X) is non-zero. Relating it back to equation (9) it means ˆ A = − B. Π

(20)

ˆ in equation (20) including the variables missing from Now substituting the value of Π the particular equation, the coefficients of equation i of the model can be given by

( X ′X )

−1

 −1  −bˆ  ( X ′Yi )  aˆi  =  i  or 0  0   

 −1 ˆ  = − ( X ′X ) bi a  i

( X ′Y )  ˆ

(21)

X ′yi = X ′yi aˆi + ( X ′X )bˆi = X ′Z iδˆi + X ′ei

(22)

δˆi = (aˆi , bˆi ) and Z i = ( yi , X i ) X ′Y = X ′yi aˆi + ( X ′X )bˆi = X ′Z iδˆi

δˆi ( ILS ) = ( X ′Z i )

−1

( X ′Y ) .

(23)

The covariance structure of X ′ei in equation (21) is σ i ,i ( X ′X ) . This need to be used to make this indirect least square estimator asymptotically efficient applying GLS procedure as (Judge et al., 1985, p.596) −1

δˆi ( ILS ) = ( X ′Z i )′ (σ i ,i X ′X ) 

−1

−1

( X ′Zi ) ( X ′Zi )′ (σ i ,i X ′X ) ( X ′Yi ) −1

 



−1

−1 −1 = ( X ′Z i )′ ( X ′X ) ( X ′Z i )  ( X ′Z i )′ ( X ′X ) ( X ′Yi )     

(24)

where σ i ,i = ( yi − ziδˆi )′ ( yi − ziδˆi )′ / T . Thus δˆi ( ILS ) exists whenever the inverse of term ( X ′Z i )′(σ i ,i X ′X )−1 ( X ′Z i )  is non-trivial; the determinant of (X′X) should be non zero. The two Stage Least Square (2SLS) estimation is a single equation method. The first stage involves using OLS of Y on X to get the predicted values of the endogenous variables, Yˆ . The second stage involves using both Yˆ and X to estimate parameters. In fact there is very little difference between the GLS method applied to the ILS above and the two stage least square method as seen from below:

Impact of exchange rate and money supply ˆ Zˆi = Yˆi , X i  =  X Π

−1 X i  =  X ( X ' X ) ( X ' Yi )  −1

369 X ( X ′X ) X ′X i  = X ( X ′X ) X ′Z i (25)  −1

−1

δˆi (2 SLS ) =  Z i′X ( X ′X ) X ′Z i  Z i′X ( X ′X ) X ′Yi = [ Z i′Z i ] Z i′Yi . −1



−1

−1



(26)

Three Stage Least Square (3SLS) method is a system-wise generalised least square technique where all equations are estimated simultaneously. It is used to correct the autocorrelation or heteroscedasticity existing in the model. Here 1 m model equations are written as (see Judge et al., 1985, p.601 for more derivations):  X ′Y1   X ′Z1  X ′Y   0 2     . = 0     .   .  X ′Ym   0

0 X ′Z 2 0

.

.

0

.

X ′Z 3 .

.

.

.

0

.

.

0   δ1   X ′e1  0   δ 2   X ′e2  0  . + .      .  .   .  X ′Z m  δ m   X ′em 

(27)

This system can be written in one equation using the Kronneker product as:

( I ⊗ X ′ ) y = ( I ⊗ X ′ ) Z δ + ( I ⊗ X ′ ) e. The covariance matrix of the error ( I ⊗ X ′ ) e is

{

(28)

∑ ⊗E ( X ′X ) . Taking account of this

δˆ(3 SLS ) = Z i′( I ⊗ X )′  ∑ −1 ⊗ ( X ′X ) −1  ( I ⊗ X ′)′Z i

}

−1

(29)

Z i′( I ⊗ X ′)′  ∑ −1 ⊗ ( X ′X ) −1  ( I ⊗ X ′)′Yi

which is distributed normally as:

(

)

−1 −1 T δˆ(3 SLS ) − δ → N  0, p lim T −1 Z ′ ∑ −1 ⊗ ( X ′X ) Z      

(30)

−1

−1 where T −1 Z ′ ∑ −1 ⊗ ( X ′X ) Z  represents the variance covariance matrix of δˆ(3 SLS ) .   This is a consistent and efficient estimator.

Appendix 3 Unit-root tests (using macro_uk.csv) The sample is 1967 (2) – 2006 (1) DLRGDP: ADF tests (T = 156, Constant; 5% = –2.88 1% = –3.47) D-lag

t-adf

beta Y_1

sigma

t-DY_lag

t-prob

AIC

3

–3.256*

0.49641

0.02237

–14.66

0.0000

–7.568

F-prob

2

–13.10**

–1.1546

0.03471

3.970

0.0001

–6.696

0.0000

1

–15.05**

–0.65147

0.03634

6.057

0.0000

–6.610

0.0000

0

–14.44**

–0.14804

0.04034

–6.408

0.0000

370

K. Bhattarai

Appendix 3 (continued) Unit-root tests (using macro_uk.csv) The sample is 1967 (2) – 2006 (1) DLM4: ADF tests (T = 156, Constant; 5% = –2.88 1% = –3.47) D-lag

t-adf

beta Y_1

sigma

t-DY_lag

t-prob

AIC

3

–2.629

0.76351

2

–4.696**

0.54428

0.01276

–7.063

0.0000

–8.691

0.01467

0.009788

0.9922

–8.419

1

–5.073**

0.54463

0.01462

0

–9.450**

0.27188

0.01574

–5.037

0.0000

F-prob 0.0000

–8.432

0.0000

–8.291

0.0000

F-prob

Tbills: ADF tests (T = 156, Constant; 5% = –2.88 1% = –3.47) D-lag

t-adf

beta Y_1

sigma

t-DY_lag

t-prob

AIC

3

–2.183

0.93751

1.086

–1.315

0.1969

0.1905

2

–2.510

0.92960

1.089

0.9859

0.3258

0.1955

0.1905

1

–2.358

0.93525

1.089

1.715

0.0883

0.1891

0.2616

0

–2.068

0.94382

1.096

0.1953

0.1342

F-prob

Exrt_dllr: ADF tests (T = 156, Constant; 5% = –2.88 1% = –3.47) D-lag

t-adf

beta Y_1

sigma

t-DY_lag

t-prob

AIC

3

–2.579

0.95813

2

–2.549

0.95873

0.07713

0.7501

0.4543

–5.093

0.07702

–0.7942

0.4283

–5.102

3.138

0.0020

1

–2.607

0.95792

0.07692

0

–2.488

0.95871

0.07910

0.4543

–5.111

0.5524

–5.061

0.0138

F-prob

CPI: ADF tests (T = 156, Constant; 5% = –2.88 1% = –3.47) D-lag

t-adf

beta Y_1

sigma

t-DY_lag

t-prob

AIC

3

–1.787

0.90181

0.8382

–5.288

0.0000

–0.3214

2

–2.778

0.83832

0.9095

–0.9907

0.3234

–0.1644

0.0000

1

–3.100*

0.82472

0.9095

–6.734

0.0000

–0.1707

0.0000

0

–5.653**

0.66898

1.032

0.07600

0.0000

F-prob

DLTbills: ADF tests (T = 156, Constant; 5% = –2.88 1% = –3.47) D-lag

t-adf

Beta Y_1

sigma

t-DY_lag

t-prob

AIC

3

–6.340**

0.086375

0.1203

–0.4912

0.6240

–4.204

2

–7.849**

0.048364

0.1200

2.489

0.0139

–4.215

0.6240

1

–7.549**

0.20553

0.1220

–1.031

0.3041

–4.188

0.0435

0

–10.91**

0.13423

0.1221

–4.194

0.0618

Impact of exchange rate and money supply

371

Appendix 3 (continued) Unit-root tests (using macro_uk.csv) The sample is 1967 (2) – 2006 (1) DLExrt_dllr: ADF tests (T = 156, Constant; 5% = –2.88 1% = –3.47) D-lag

t-adf

beta Y_1

sigma

t-DY_lag

t-prob

AIC

3

–5.563**

0.23968

0.04347

–0.2538

0.8000

–6.240

2

–6.436**

0.22356

0.04334

–0.6590

0.5109

–6.252

1

–8.173**

0.17964

0.04326

0.7509

0.4539

0

–9.855**

0.22652

0.04319

F-prob 0.8000

–6.262

0.7806

–6.271

0.7881

The above-mentioned ADF tests suggest that the growth rate of GDP (DLRGDP), growth rate of money supply (DLM4), inflation (CPI) were stationary. Treasury bills rate (Tbills) and exchange rate (Exrt_dllr) were non-stationary but the percentage changes in the treasury bills rate (DLTbills) and exchange rates (DLExrt_dllr) were stationary.