Economic and Social Review, Vol 10, No. 2, January, 1979, pp. 147—156
T h e I r i s h Aggregate I m p o r t D e m a n d
E q u a t i o n : the ' O p t i m a l ' F u n c t i o n a l F o r m
T . B O Y L A N * M . C U D D Y
I . O M U I R C H E A R T A I G H
University College, Galway
Precis: A more general approach than that of "best fit" for choosing empirically the appropriate
formulation of the aggregate import demand function for Ireland is presented. This approach leads to the choice of a particular form, from a class of forms, for a given specification of the aggregate import demand function. It is found that the log linear form is preferred to the linear form where gross national income and the domestic to foreign price ratio are the explanatory variables. It is also found that the inclusion of a partial adjustment mechanism does not significantly improve the model specification.
I I N T R O D U C T I O N
I
n studies o f aggregate i m p o r t demand ( K r e i n i n , 1 9 6 7 ; H o u t h a k k e r and Magee, 1 9 6 9 ; K h a n , 1 9 7 4 ; and Magee, 1 9 7 5 ) , t w o f u n c t i o n a l forms have p r i n c i p a l l y been used:(i) a linear f o r m u l a t i o n i n w h i c h i m p o r t s are assumed t o be a linear f u n c t i o n o f the explanatory variables selected, and
(ii) a l o g linear f o r m u l a t i o n i n w h i c h the l o g a r i t h m o f i m p o r t s demanded is assumed t o be a linear f u n c t i o n o f the logarithms o f the explanatory variables.
A r e c u r r i n g p r o b l e m encountered i n the literature is the choice o f the appropriate f u n c t i o n a l f o r m f r o m w i t h i n the restricted class o f linear and l o g a r i t h m i c functions. The choice is made d i f f i c u l t and u l t i m a t e l y quite
*The authors wish to thank the referees for their helpful comments on an earlier draft.
arbitrary for t w o reasons. First, at the theoretical level, there are n o a priori economic criteria to indicate that one f u n c t i o n a l f o r m is superior t o the other. Secondly, at the empirical level, d i s c r i m i n a t i o n between f u n c t i o n a l forms based o n certain "goodness o f f i t " criteria, p r i n c i p a l l y the coefficient o f d e t e r m i n a t i o n ( R2) , can be quite arbitrary. There are b o t h economic and
statistical implications o f using one f o r m o f the equation rather t h a n an other. T h e use o f a linear f u n c t i o n a l f o r m , for instance, implies a decreasing price elasticity o f i m p o r t demand and an income elasticity tending t o w a r d u n i t y . While the use o f a logarithmic f o r m u l a t i o n implies constant elasticities w i t h respect t o price and income, this may be considered theoretically t o o restrictive i n the case o f an i m p o r t f u n c t i o n . Statistically, mis-specification o f the functional f o r m results i n the v i o l a t i o n o f the classical properties for the error term. This results i n the estimates being biased and inconsistent ( K m e n t a , 1 9 7 1 ) , and thus weakens its predictive power.
Previous studies o f Irish i m p o r t functions b y Leser ( 1 9 6 7 ) , Baker and D u r k a n ( 1 9 6 9 ) , and McAleese( 1970a, 1970b) have all used linear or log linear or lagged variants o f either o f these t w o forms for estimation purposes. McAleese ( 1 9 7 0 b , p . 399) identifies the p r o b l e m o f choosing between the t w o f u n c t i o n a l forms and the i n a b i l i t y o f researchers to select the appro priate f u n c t i o n a l f o r m because o f the absence o f appropriate selective criteria.
The purpose o f this paper is t o present a more general approach t h a n that o f "best f i t " for choosing empirically the appropriate f o r m u l a t i o n o f the aggregate i m p o r t demand f u n c t i o n for Ireland. The procedure is based o n the w o r k o f B o x and Cox ( 1 9 6 4 ) . A generalised f u n c t i o n a l f o r m o f the i m p o r t demand f u n c t i o n is specified. This f o r m includes as special cases b o t h the linear and the l o g linear forms and also includes a w h o l e range o f other possible forms. The Box-Cox procedure consists essentially o f deter m i n i n g w h i c h member o f this class is o p t i m a l i n a certain sense ( t o be defined) and also i n determining whether or n o t a specific f o r m w i t h i n the class (e.g., linear) is acceptable i n relation t o the given observations.
The l a y o u t o f the paper is as follows. I n Section I I the aggregate i m p o r t demand equation i n linear and l o g linear f o r m is specified. B o t h these equations are m o d i f i e d t o include a partial adjustment mechanism. The generalised f u n c t i o n a l f o r m o f the i m p o r t demand equation is t h e n defined along w i t h the m a x i m u m l i k e l i h o o d m e t h o d o f estimating its parameters. T h e empirical results are presented i n Section I I I and the summary and conclusion i n Section I V .
I I T H E G E N E R A L I S E D F O R M O F T H E A G G R E G A T E I M P O R T D E M A N D F U N C T I O N
prices and t o the level o f domestic real i n c o m e .1 This gives
Md = F (P,Y) (1)
where
Md = the q u a n t i t y o f i m p o r t s demanded,
P = the r a t i o o f the price o f i m p o r t s t o the domestic price level, and Y = the real Gross N a t i o n a l Product.
T h e sign o f the partial derivative, 5 Md/ 5 P , is expected t o be negative, w h i l e
the sign o f 5 Md/ 5 Y is generally expected t o be p o s i t i v e .2 T h e linear f o r m u l
a t i o n o f the aggregate i m p o r t equation for t i m e t is
Md = aQ + a , Pt + a2Yt + et (2)
where e is assumed t o be a r a n d o m error t e r m . I f a l o g a r i t h m i c relationship is considered preferable, then the aggregate demand for the i m p o r t s for t i m e t is
log M d = 0O + ftlog Pt + h log Yt + et (3)
Equations (2) and ( 3 ) , as f o r m u l a t e d , are ex ante relationships and the replacement o f i m p o r t demand, Md, b y actual i m p o r t s implies instantaneous
adjustment t o changes i n relative prices and real i n c o m e . T h i s , however, is regarded as an excessively restrictive assumption and can be relaxed b y specifying a partial adjustment mechanism for i m p o r t s i n w h i c h the change i n i m p o r t s is related t o the difference between the ex ante demand for i m p o r t s i n p e r i o d t and the actual level o f i m p o r t s i n the previous p e r i o d . For the linear f o r m , this m o d e l reduces t o
Mt = aQ * + ax *Pt + a2 * Y( + a3 * Mt_ l + et (4)
S i m i l a r l y , for the l o g a r i t h m i c f o r m , the partial adjustment mechanism yields
log Mt = /30 * + ^ * l o g Pt + 02 *log Yt + /33log Mt_ ! + et (5)
I n this paper, equations specified as i n (2) and (3) are termed M o d e l I , and equations specified as i n (4) and ( 5 ) , w h i c h c o n t a i n the partial adjust m e n t mechanism, are termed M o d e l I I . A p p l y i n g a p o w e r t r a n s f o r m a t i o n t o each o f the variables and w r i t i n g M j1 = Mt for n o t a t i o n a l convenience,
1. See Learner and Stern (1970, ch. 1).
the generalised f u n c t i o n a l f o r m i n the case o f e q u i l i b r i u m i m p o r t demand ( M o d e l I ) is
Mx - 1 = + P* _ 1 Y * — 1 /f ix
To + T i _ t + y _ t + e, \v)
X X X
while for dynamic i m p o r t demand (Model I I ) the f o r m is
Mx — 1 Px — 1 YK — 1 Mx — 1
X X X X
F o r X = 1, equations (6) and (7) become identical t o the linear specifications (2) and ( 4 ) . F o r X = 0, equations (6) and (7) reduce t o the l o g linear equations (3) and ( 5 ) . I t should be n o t e d that for X = 0, the expressions i n v o l v i n g X appear t o become indeterminate. However, i f we expand, say, the transformed dependant variable, we o b t a i n
M * - 1 = el o§ Mt - l _ eX I o§M t - l
X X X
= ~ { l + Xlog Mt + V6(Xlog Mt)2 + - 1 }
= l o g Mt + ^ ( l o g Mt)2 +
M * - 1
For X = 0, = l o g Mt and similarly for the other variables.
X
The Box-Cox procedure as applied t o M o d e l I is presented here. A p p l i c a t i o n o f the procedure t o M o d e l I I involves, i n effect, merely the inclusion o f an a d d i t i o n a l explanatory variable and does n o t materially affect the m e t h o d . For n o t a t i o n a l convenience equation (6) is r e w r i t t e n as
Mt( X ) = To + T j Pt( X ) + T2 Yt( X ) + et = nt (X) + et (8)
The p r o b a b i l i t y density o f the untransformed observations, Mt, and hence
the l i k e l i h o d in relation to these original observations is obtained b y m u l t i p l y ing this n o r m a l density b y the Jacobian J ( X ; M ) o f the t r a n s f o r m a t i o n . I n our case
J ( X ; M ) = n M ^ -1 (9)
and, hence, the l i k e l i h o o d given the original observations is
L (7 o,T l,7 2, a * , X / M ) = A . exp ^ - 2 ( Mt( X ) (10)
T h e m a x i m u m l i k e l i h o o d e s t i m a t i o n o f the u n k n o w n parameters (7Q> 7J >
72> °2 > A) is a two-stage procedure. First, f o r given X, equation (10) gives
the l i k e l i h o o d for a standard least squares p r o b l e m apart f r o m a constant factor. Hence, the m a x i m u m l i k e l i h o o d estimates o f the 7's are the least squares estimates for the regression p r o b l e m w i t h dependent variable Mt'x'
and explanatory variables P1^ ' and Y^x\ and the m a x i m u m l i k e l i h o o d
estimate o f a2, denoted for f i x e d X b y ff2 (X), is
a2( X ) =s 2 (X) (11)
n
where S2 (X) is the residual sum o f squares i n the analysis o f variance o f
M /x> . Hence for f i x e d X, the m a x i m i s e d l o g l i k e l i h o o d is, except f o r a
constant factor,
L m a x W = - M m l o g a2 (X) + ( X - l ) E l o g Mt (12)
B y p e r f o r m i n g the above analysis o n the transformed observations, Mt( X ) ,
for a t r i a l series o f values o f X, i t is possible t o p l o t Lm a x( X ) against X and
f r o m this p l o t t o read o f f the estimated value o f X f o r w h i c h L (X) is
r max v '
itself m a x i m i s e d w i t h respect t o X. Based o n the o r t h o d o x large sample t h e o r y o f m a x i m i s e d l i k e l i h o o d estimation, i t can be shown t h a t a 100 (1 — a) per cent confidence interval for X is given b y values o f X such that
Lm a x W - Lmax( X ) <1/2X2( a ) (13)
F i n a l l y , this confidence interval enables us to examine the acceptability or otherwise o f any hypothesised value o f X (e.g., i n particular X = 0 (log linear) or X = 1 (linear)).
T h e B o x - C o x procedure as applied here consists o f considering the generalised f u n c t i o n a l forms o f equations (6) and (7) f o r a series o f values o f X, o b t a i n i n g the m a x i m u m l i k e l i h o o d estimates o f the parameters o f the transformed m o d e l f o r each such X, f i n d i n g the value o f X f o r w h i c h the log l i k e l i h o o d i n r e l a t i o n t o the original observations is m a x i m i s e d , and finally e x a m i n i n g i n particular the p o s i t i o n o f the linear and l o g linear models w i t h i n this class o f functions.
I l l E M P I R I C A L R E S U L T S
A n n u a l data for the p e r i o d 1953 t o 1975 inclusive are used i n this study. Real i m p o r t s i n year t , Mt, are the value o f i m p o r t e d goods and services at
constant 1970 prices. Real Gross N a t i o n a l Product (GNP) i n year t , Yt, is
to
T a b l e 1: Parameter estimates, t-valuesa, R s and the d statistics for the linear and log linear forms of Models I and I I
Model
Dependent
Variable Constant
Pt
LogPt
Yt
LogYt
Mt - 1
Log Mt_j R2 d
I - 4 2 3 . 6 2 (.- 1 1 . 1 9 )
- 8 3 . 9 6 ( - 1.05)
0 . 6 9 ( 2 9 . 4 7 )
. 9 8 7 8 2.15
L o g Mt - 6 . 7 5
( - 1 5 . 6 9 )
- 0 . 5 2 ( - 3 . 3 7 )
1.79 ( 3 0 . 7 6 )
. 9 9 1 0 2 . 0 2
I I Mt - 3 9 0 . 8 8
( - 2 . 6 1 ) - 1 1 5 . 1 0 ( - 0 . 9 2 )
0 . 6 5 ( 3 . 4 2 )
0 . 0 6 ( 0 . 2 4 )
. 9 8 7 6 0 . 2 6
L o g Mt - 4 . 9 9
( - 3 . 0 9 )
- 0 . 6 0 ( - 3 . 6 0 )
1.37 ( 3 . 6 2 )
0 . 2 2 ( 1 . 1 5 )
. 9 9 1 3 1.02
w n o O 2 B > 2 a O n > r w < S
o f the i m p o r t price index (base 1970) t o the Wholesale Price I n d e x (base 1 9 7 0 ) , b o t h i n year t . The Wholesale Price I n d e x series is taken f r o m the Irish Statistical B u l l e t i n (various issues). A l l other series are taken f r o m the D a t a Bank o f A n n u a l Economic T i m e Series, 1977 o f the Central Bank o f Ireland's Research D e p a r t m e n t .
First, certain statistics associated w i t h equations (2) and (3) (the linear and l o g linear forms o f M o d e l I ) and equations (4) and (5) (the linear and l o g linear forms o f M o d e l I I ) are estimated. These include the parameter estimates and their respective t-values, the coefficient o f d e t e r m i n a t i o n ( R2) and the Durban-Watson statistic f o r a u t o c o r r e l a t i o n ( d ) . These statistics
are presented i n Table 1.
The results f r o m M o d e l I show that the parameter estimates have the signs expected f r o m t h e o r y i n b o t h the linear and l o g linear case. T h e parameter estimate for income has about the same degree o f significance for b o t h the linear and l o g linear f o r m . A t the 5 per cent significance level the parameter estimate for the price variable is n o t significant f o r the linear f o r m , b u t is significant for the l o g linear f o r m . B o t h the linear and l o g linear f o r m give a very high R2 and a d statistic w h i c h allows acceptance o f the hypothesis
t h a t there is no a u t o c o r r e l a t i o n . Hence, b y reference t o these criteria i t is d i f f i c u l t t o discriminate between these t w o forms o f specification.
S i m i l a r l y , i n the case o f M o d e l I I the signs o f the parameter estimates for b o t h forms o f the equation are as expected. A g a i n , as i n M o d e l I , the income parameter estimate has a p p r o x i m a t e l y the same level o f significance under b o t h forms while the price variable parameter estimate is n o t significant under the linear f o r m , b u t is significant under the l o g linear f o r m . The parameter estimate for the lagged variable is n o t significant under either f o r m . A g a i n , the m o d e l seems t o be w e l l specified under either f o r m , given the magnitude o f the R2. Whereas the hypothesis o f non-autocorrelation must
be rejected under the linear f o r m , under the l o g linear f o r m the d statistic lies i n the inconclusive region. However, recognising that the d statistic is inappropriate i n the case o f lagged models, an alternative test ( D u r b i n , 1970) was carried o u t w h i c h allowed acceptance o f the hypothesis that there is n o a u t o c o r r e l a t i o n i n either the linear or l o g linear f o r m o f M o d e l I I . O n the basis o f these results, one thus cannot j u s t i f y the choice o f one f o r m rather t h a n the other i n the case o f either M o d e l I o r I I .
Lr a a x( A ) is calculated f o r Models I and I I for values o f X f r o m —1.5 t o
+1.5 at intervals o f 0 . 1 . This series o f values includes X = 0 and X = 1. A 95 per cent confidence interval for X is derived as explained i n Section I I . Lm a x( X ) p l o t t e d against X and the 95 per cent confidence interval for X is
shown i n Figure 1. I t may be observed that Lm a x( X ) i n the range examined
The m a x i m u m value o f Lm a x( X ) for M o d e l I occurs at X = 0.24. T h e 95
per cent confidence interval for X is (—0.14, 0.62). This includes X = 0 and does n o t include X = 1. The hypothesis is t h a t X = 0 cannot be rejected whereas the hypothesis is t h a t X = 1 can be rejected. I t m a y , therefore, be c o n c l u d e d that the l o g linear f o r m u l a t i o n o f the i m p o r t demand f u n c t i o n as specified i n M o d e l I is superior t o a linear f o r m u l a t i o n . The m a x i m u m value o f Lmax (X)
for M o d e l I I occurs also at X = 0.24 and the 95 per cent confidence i n t e r v a l for X is (—0.15, 0 . 6 2 ) . As i n the case o f M o d e l I , this includes X = 0 and ex cludes X = 1. Thus, i n the case o f M o d e l I I also, the l o g linear f o r m u l a t i o n is superior t o the linear f o r m u l a t i o n .
The i m p o r t demand elasticity w i t h respect to price and income is i m p o r t a n t for longer-term forecasting. I t is o f interest t o compare the elasticities derived i n this study w i t h those o f previous studies. T h e elasticities derived f r o m the l o g linear f o r m i n this study are the more appropriate t o compare w i t h elasticities f r o m other studies, since the l o g linear f o r m was f o u n d t o be acceptably close t o the o p t i m a l f u n c t i o n a l f o r m . I t m i g h t be n o t e d that the income and price elasticities derived f r o m the o p t i m a l f u n c t i o n a l f o r m are b o t h very close t o those derived f r o m the l o g linear f o r m i n b o t h Models I and I I .
The price elasticity derived f r o m Models I and I I are b e l o w the lowest value i n McAleese's (1970a) range and considerably b e l o w Leser's ( 1 9 6 7 ) value (Table 2 ) . S i m i l a r l y , the income elasticities derived f r o m Models I and I I are b e l o w the lowest value o f McAleese's while that o f M o d e l I is greater, and t h a t o f M o d e l I I less, t h a n the value derived b y Leser.
T a b l e 2: The price and income elasticities of import demand
Log Linear
Elasticity Model I Model I I McAleese Leser
Price - 0 . 5 6 - 0 . 6 8 * - 0 . 8 9 - - 1 . 5 3 - 1 . 3 8 I n c o m e 1.79 1.56 1 . 8 7 - 2 . 1 5 1.61
* L o n g - r u n elasticities
I V C O N C L U S I O N S
This paper is concerned w i t h finding the appropriate f o r m for an aggregate i m p o r t demand f u n c t i o n for Ireland and, i n particular, d i s c r i m i n a t i n g between the linear and l o g linear f o r m u l a t i o n s o f a standard specification. T h e absence o f theoretical grounds for choosing a particular f o r m u l a t i o n necessitated an e m p i r i c a l approach.
-u l a t i o n i n b o t h a static and a p a r t i a l adj-ustment m o d e l . The p a r t i a l adj-ustment m o d e l added n o t h i n g t o that o f the static m o d e l . The l o g linear f o r m u l a t i o n of the static m o d e l thus seems the more appropriate f o r m and specification of the aggregate i m p o r t demand f u n c t i o n for Ireland.
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