- Email: [email protected]
On the determination of regional base and regional base multipliers
Regional
Science
and Urban
Economics
22 (1992) 619-635.
North-Holland
On the determination of regional and regional base multipliers*
base
Scott J. Brown Pat@
First Bank, Seattle WA, USA
N. Edward
Coulson
The Pennsylvania
State Unioersity, University Park PA, USA
Robert
F. Engle
University of California at San Diego, La Jolla CA, USA Received
August
1989, final version
received
February
1992
Given that employment in a region’s economic base and total employment are in long-run equilibrium, the two series ought to be cointegrated. This suggests a method for discovering the region’s basic sectors which is explored using the Philadelphia MSA as an example.
1. Introduction
The base multiplier is one of the oldest, simplest, most intuitive and practical models of regional aggregate economic behavior in regional science. For discussions of the merits and faults of this model, there are recent surveys by Richardson (1985) and Nijkamp et al. (1986). Both of these point out that the most difficult and important problem in base analysis is that of defining the base - that is, that portion of regional supply which is devoted to exports. Methods of base identification which have been devised sometimes lack theoretical justification and none seem to be derived from the properties of the multiplier model itself. The method proposed here does not have this defect. In what follows the fact that the base multiplier model is a model of long-run equilibrium is used to yield a stationarity condition on the residual of the regression equation used to estimate the base multiplier. Then basic and total earnings are said to be cointegrated, and so cointegrating tests can be used to test whether the definition of the base is correct. In Correspondence to: ings, The Pennsylvania *We would like to helpful comments, and 016&0462/92/$05.00
Edward Coulson, Department of Economics, 608 Kern State University, University Park, PA 16802, USA. thank Sam Yoo, Mark Watson, and the anonymous Mark Hutchens for research assistance.
0
1992-Elsevier
Science Publishers
Graduate referees
B.V. All rights reserved
Buildfor very
S.J. Brown et al., The determination of regional base
620
section 2, the base multiplier is examined and the test is outlined. In section 3 the problem of base identification is explored and some difficulties of the test examined, followed in section 4 by an empirical example using data from the Philadelphia metropolitan area. Section 5 concludes. 2. The base multiplier model The simple mechanics of the base multiplier model are well known to every regional economist. Some regional aggregate - income, employment and earnings are three examples - is divided into two components: that portion which is exogenously generated by export demand, and that which is endogenously generated by local demand. The first component, which is also called basic demand, provides injections of income into the regional economy, which induces further local spending and a subsequent multiplicative effect in the standard Keynesian fashion. For specificity, and because the empirical example below deals with it, let the aggregate measure of interest be regional employment (E). Then we can write
(1)
E=B+L,
where B is regional is locally generated behavioral equation
employment generated by basic (export) demand and L employment. We can add to this definitional equation a analogous to the Keynesian consumption function:
C=a+bY,
(2)
where in this case Y is total regional income and C is final demand for local goods and services. In this scenario, b is the marginal propensity to consume locally (i.e., neither save nor import from other regions) and a, which is often represents autonomous local demand. suppressed in multiplier studies, Letting 8 be labor’s share of output divided by average wages, multiplying through eq. (2) by this parameter yields L=aO+bE,
(3)
since E = 8Y and L= BC by our definitions. substituting we have the reduced form’ ad E=(l-b)+(l-b)B’
Appropriately
rearranging
and
1
‘Because we assume the region acts as a small open economy earnings can be assumed to be exogenous, though this is not critical below.
within its nation, export for the cointegration tests
S.J. Brown et al., The determination of regional base
621
so that l/( 1 -b) is the (marginal) base multiplier which gives the increase in total regional employment given a unit increase in B, and is the parameter of interest here. This model is quite simple, and it can be extended in any number of ways. Nevertheless, it is a good prototype, in particular since it yields a constant marginal multiplier which seems a reasonable approximation and is often used in practical applications by regional analysts. The practicality of the base multiplier model is its most obvious attribute [see Nijkamp et al. (1986)]. The question then is, how is one to estimate the base multiplier for a specific region? One obvious first step is to treat (4) as a regression equation, i.e., rewrite it (adding an error term) as E=cr+flB+o,
(5)
and then gather (for the particular region) a time series of observations on E and B, run the regression and obtain estimates of the multiplier, p. However, this leaves aside two questions of major importance: (a) How is B calculated? and (b) What will the properties of the error term be? With regard to (a), which, as noted, is the major issue in the application of the base multiplier model, the ultimate, though very expensive answer, would be to sample a number of firms, presumably stratified by Standard Industrial Classification (SIC) code, and apply the proportion of sales the sampled firms give as export-oriented to the aggregate employment of that sector, then adding up the export employment from the various sectors to obtain B. However, this will obviously be very costly, and likely to be infeasible for many researchers. Richardson (1985) examines several non-survey methods of obtaining estimates of B. The simplest and crudest of these is the assignment method, wherein the earnings within broad employment categories are wholly assigned to either basic or local earnings. Often, manufacturing employment is assigned to the base and all other employment is assumed to be locally generated. While this seems particularly crude, it should be noted that this assumption underlies the structural econometric models of Glickman (1977) insofar as such models contain equations where manufacturing output (and only manufacturing output) is solely a function of macroeconomic variables. Other methods of base determination do not make the assumption that an industry’s employment must be entirely basic or local, but rather use some secondary source of information to determine the proportion of employment to be assigned to each. The secondary source is often national employment in the industry, which can be used to calculate location quotients.’ However users of either the assignment method or any of the suggested *See also Mathur
and Rosen (1974).
622
S.J. Brown et al., The determination of regional base
methods of division have no way of checking the adequacy of their base measurement, so that (as has been noted) estimation of the base multiplier is likely to be highly dependent on what are, of necessity, ad hoc assumptions about the nature of the base. This brings us to the second of the two difficulties mentioned above. Supposing that the problem of base identification has been solved, what are the consequences of estimating (5) in a time-series framework? Such a regression is indeed fraught with difficulties, the major one being that it is quite probable that both E and B are non-stationary series - that is, they exhibit stochastic trends. Certainly national aggregates follow such trends as the results of Nelson and Plosser (1982) indicate. Therefore at least some of the component regional aggregates must do so as well. The difficulties of estimation and inference in regression equations such as (5) when one nonstationary series is regressed on another have been well-documented in Granger and Newbold (1974) who call such regressions ‘spurious’. They note that two series might be independent random walks (perhaps with drift) having no other relationship whatsoever, and yet when one is regressed on another, the resulting R2 and t-statistic can be extremely high, and lead the investigator to conclude that a ‘good lit’ has been obtained. The danger for base multiplier estimation is clear: Any base calculation, correctly or incorrectly done, will appear to lit well with the data and perhaps lead one to assume that the true base has indeed been found, and that the base multiplier has been consistently and efficiently estimated. This danger is detectable because the properties of the residual series in spurious regressions will certainly not conform to the ideal conditions. In particular, since linear combinations of non-stationary series are typically non-stationary themselves, the particular linear combination E, - &- fill, = il, will have this property. Thus the residuals will be non-stationary and so have infinite variance and undefined mean (in population) which hardly lends itself to accurate inference. On another level, if the residual of the regression has an infinite variance, then it would seem that E and B (or rather a linear function of B) can drift infinitely far apart, which would suggest that B and E in fact do not have an (long-run) equilibrium, which is certainly a contradiction of the idea underlying the base multiplier model. However Engle and Granger (1987) have pointed out that it need not be the case that linear combinations of non-stationary variables are themselves non-stationary. Certain combinations may be such that the non-stationary parts ‘cancel each other out’ leaving only stationary residuals. If such a linear combination exists, the two variables are said to be cointegrated and the existence of this possibility has crucial ramifications for our problem. For consider that if 6,, the estimated residuals, are stationary, then they have finite variance, with the attendant implication that E, and B, will drift infinitely far apart only with infinitely small probability. Engle and Granger
S.J. Brown et al., The determination of regional base
623
(1987) note that this provides an operational definition for (long-run) equilibrium in an econometric context, for if two or more variables are in long-run equilibrium, then even if there are continual random shocks to the equilibrium system the variables should move together over time. This will be the case if u, is stationary, and will not if U, is non-stationary. With the presumption that the base multiplier model is indeed a model of long-run equilibrium between E and B, the above discussion contains a potentially powerful suggestion for the problem of base determination. B can be necessarily defined as the set of earnings flows which cointegrates with E in estimation of (5). Thus we can undertake a search process, estimating eq. (5) under OLS using several different definitions of the base, and test the residuals for stationarity.3 If the residuals are non-stationary, then E and B do not cointegrate, they can presumably drift infinitely far apart, and they do not form the long-run equilibrium. Thus the given definition of B cannot be the true definition of the base. By contrast, stationarity in the residuals would indicate that the true B has been found. Notice also that a consistent estimate of the multiplier itself has also been found in the estimate of (5). In fact Stock (1987) suggests that this estimate is ‘super-consistent’ since the estimator converges to a faster rate than standard OLS estimators. However it turns out that the t-statistics may be far from the standard distributions, particularly in small samples. The short-run dynamics should be modeled, presumably in an error correction framework as suggested by Engle and Granger (1987). Special cases of the error correction mechanism, such as the partial adjustment model, have been previously used in this context [Sasaki (1964)].
3. Cointegration
and problem of base identification
In this section we wish to expand the discussion of the preceding section. More particularly, the connection between stationarity of the residuals in the cointegrating regression (5) and the identification of the base is subject to two difficulties, which correspond to the standard type I and type II errors: (1) Is it possible to reject a base definition when it is correct? and (2) Is it possible to accept a base definition when it is wrong? As we shall see in the next section where we discuss estimation issues, the test for the stationarity of the residuals has as its null hypothesis the existence of a unit root in the lag polynomial which models the residual
‘Such a search process raises the question of be, which we ignore in our empirical example. correlated (even if it is stationary) so that the only in the long run. Thus we can only describe
what the nominal sizes of successive tests should It should be noted that v, will be highly serially variables will return to the equilibrium position E = a + /3B as a long-run equilibrium.
S.J. Brown et al., The determination of regional base
624
series. Thus the null hypothesis is one of non-stationarity. While this is a non-standard testing problem, Engle and Granger (1987) and others have developed tests, the Monte Carlo simulations of which generate appropriate critical values for tests of standard sizes. In attempting to answer (l), the appropriate question is whether it is possible to accept the null hypothesis of non-stationarity when in fact the residuals are stationary. Thus the power of the tests become important. While tests for non-stationary residuals are well known to have low power for close alternatives, the simulation results of Engle and Granger show that some of the tests have reasonable power against alternatives which are not too close. The importance of power against close alternatives depends on the importance placed on the distinction between an equilibrium which takes a very long time to achieve, and a dynamic model which has no equilibrium. In practice this distinction may not be observable in a finite amount of data [Nelson and Plosser (1982)], and may not be important. While the issues surrounding question (1) are fairly clearcut (even if the answer is not) the analysis of question (2) is somewhat more subtle. The idea is that while a correct base definition is sufficient to yield a stationary residual, it is not necessary if there are other cointegrating relationships among the variables in the base multiplier model. Since arbitrary finite linear combinations of stationary variables will yield stationary variables in turn, if the base multiplier model (5) replicates a linear combination of other cointegrating equations, then a stationary error would result, even if the base definition used is incorrect. To see this, consider a simple region with four industries, with employment xi,i=l,..., 4. Let X, and X, be basic sectors so that B= X, +XZ and
(1 -P)W, +X,)+X,+X,=0,, where v, is a stationary sequence.4 The above equation is merely a reformulation of (5) for this simple scenario. If the investigator mistakenly identifies the base as X, +X,+X, and (6) is the only linear combination of the earnings variables which yields a stationary residual, then by forcing X, to have a coefficient of (1 -p) rather than one (noting /3#0) the investigator must automatically observe a non-stationary residual and would thereupon recognize that the base has not been correctly identified. However, (6) may not be the sole cointegrating relationship. For example, the two local sectors may be growing together over time; that is x4--Y,x,=o,:
41n what follows the intercept estimated regressions below.
term is suppressed
for simplicity,
though
it is included
in the
625
S.J. Brown et al., The determination of regional base
may be a cointegrating incorrect base multiplier
equation mode1
with
(l-p*)(x,+x~+x,)+x,=w, is estimated. OLS will search for that and so will search for cointegrating yield infinite variances (in population). In this case the answer is yes. Let zi and (7) and solve the following set of a free parameter
v2 stationary.5
Now
suppose
the
(8) fl* which minimizes the variance of w,,, vectors in preference to vectors which Does such a linear combination exist? and z2 be the weights given to eqs. (6) linear equations, noting that /I* is also
z,(l -P)=(l-P*)>
(94
zt(l -P)=(l -P*),
(9’4
z,-Yz2=(1-B*),
(9c)
z,+z,=l.
(9d)
Since (9a) and (9b) are dependent equations we are left with three equations and three unknowns which have the solution zr = y//3+ y, z2 =/?//I+ y and /I* =( 1+7)/I/(/I + y). Thus, if (8) is estimated instead of (6), the OLS estimation method will pick off the given /I* because this will yield the stationary residual we = (y/j? + y)v, +(/I/j? + y)v,_ Even though a stationary residual has been estimated, the base has not been correctly identified, and noting that /I > 1, /I* < fi. That is, the multiplier has been underestimated, which of course is a standard result when too much has been included in the base. Consider, on the other hand, the possibility of including too little in the base. Suppose the investigator estimates
x,(1-~*)+x,+x,+x,=w”,
(10)
in which case we have
z,(l -B)=(l-8*), z,(l-B)=l, z1
-yzz
= 1,
*The suggestion is based on the presumption that supply and demand for the region’s locally produced services are not subject to permanent taste or productivity shocks so that the ratio of earnings stays (approximately) constant over time.
S.J. Brown et al., The determination of regional base
626
Zl
+z, = 1,
and it is easy to see that (even aside from the fact that we have four equations and three unknowns) unless fi* =y =O, these equations have no solution. However, we are being somewhat too restrictive here. We are allowed to postulate up to three cointegrating relationships since we have four variables in the system: Thus along with allowing the two local sectors to grow together over time, let us allow the two basic sectors to do so as we11.6 We add
(11)
x1-ljx2=ug to the system of structural equations, with uj stationary. a staionary residual residual if there is a solution to
z,(l -P)-z,= z1=
1,
z1=
1.
Then (10) will have
1,
(Note that z2 must equal zero here.) The solution is clearly zr = 1, zs =/3 yielding p* =fi( 1+$), and since $ >O, /I* >/I, the standard result when too little is included in the base. Examples with greater numbers of basic and local sectors can also be constructed; the analysis becomes more complex because a richer variety of cointegrating relationships can be analyzed. The general conclusions to be derived from such analysis are: (1) false acceptance of too small a base is impossible when basic sectors are not cointegrated; (2) false acceptance of too large a base is impossible when local sectors are not cointegrated; (3) false acceptance of a base is more likely, the larger the rank of the cointegrating matrix, i.e., the larger the number of cointegrating relationships. (1) and (2) can be seen from expanding the above four-industry analysis to an arbitrarily larger number. (3) arises from the fact that when there are fewer cointegrating relationships among (say) the basic sectors, it is obviously more difficult to replicate a linear combination of them in any given cointegrating regression. 6As in footnote 5, the possibility that the two export sectors are cointegrated is based on the absence of permanent taste or productivity shocks. If it is the case that such shocks are indeed present so that secondary cointegrating relationships do not exist, then (6) is the only cointegrating equation and the stationarity of the residual is suflicient to define the base.
S.J. Brown et al., The determination of regional base
627
We now address the question of the division of sector earnings into its basic and local components. Letting pi be the portion of the ith sector devoted to exports, the base multiplier model can be written as
but obviously the model as it stands is not identified. A least squares projection would simply give back $J= 1 for all i. Some prior information on the 4i as derived, for example, from location quotients, is necessary. The advantage of the cointegration technique described here is that if the prior information is wrong, the regression will return to a non-stationary error, and the researcher will know that the true base definition has not been found. Secondly, there is the issue of allowing differential multipliers across industries. Here the model would be
and the identification problem is obviously more severe. The amount of prior information necessary depends on the variability of the pi. In the extreme case where each sector’s multiplier is separately estimated, each 4i must be calculated beforehand; we then have the reduced form of the standard inputoutput model. 4. Estimation The specification of the base in a base multiplier model must necessarily proceed in three steps. First, it must be ascertained that the employment series under consideration (total employment and whatever passes for basic employment) are indeed non-stationary processes. Second, the various detinitions of export base must be calculated. Third, cointegration tests must be performed, to see if the residuals are stationary, as required by multiplier theory. The tests of stationarity required in the first and third parts (and as it turns out, the second as well) are generally assumed to be first order tests, that is, whether the variable of interest is a random walk (perhaps with drift) or not. Therefore, in the language of classical statistics, the test is of H,: p = 1 in the regression 4t= P4t - 1+ 4,
for some variable qt. If He is rejected in favor of the alternative p < 1, then qt is a stationary process. If not, the random walk null is not rejected. Of the several testing procedures, the one which has found the most favor in the
628
S.J. Brown et al., The determination of regional base
literature is the Dickey-Fuller regression to
test, which involves
the transformation
of the
(12) so that the null is transformed into a zero restriction on the coefftcient of the lagged dependent variable. Then the ordinary t-ratio given in regression output can be used as a test statistic. In the case here, this t-ratio is not distributed as a t, nor is it asymptotically normal, but Fuller (1976) gives critical values based on Monte Carlo simulation. The case above describes the test when the null does not allow drift, and the alternative is a zero-mean stationary process. This is appropriate when q1 is a residual, as would be the case in the cointegration tests. In fact, the cointegrating case is complicated by the fact that the distribution of the Dickey-Fuller test statistic widens as the number of variables in the cointegrating equation increases above one. However Engle and Yoo (1987) provide appropriate critical values for the cointegration case. Testing for the non-stationarity of a raw employment series is complicated by the fact that such series may have drift. Hence an intercept term may be added to (12) above. With a non-zero intercept under the alternative, the series is seen to have a non-zero long-run equilibrium. West (1988) has shown that the slope coefficient is asymptotically normal in this case, even under the null. On the other hand, the simulated Dickey-Fuller critical values for this case appear to be far from standard. Investigations by Hylleberg and Mizon (1989) indicate that the normal distribution is not a good approximation in finite samples unless the drift is large (like 100 times the variance of u,). This does not appear to be the case. We will refer to both normal and Dickey-Fuller results below. Before turning to the estimation, a word about the data is in order. The sample consists of total and sectoral employment at monthly intervals for the Philadelphia Metropolitan Statistical Area from January 197551987. The actual sample period within which the multiplier is estimated is cut off at December 1986 in order to facilitate post-sample comparisons. The sectoral employment series are generated from one and two-digit SIC employment data from Employment and Earnings: States and Areas. The sectors considered are given in column 1 of table 1. In column 2 the Dickey-Fuller test statistic derived from estimation of (12) and including a drift term is given for each of these sectors’ raw employment series. Asterisks there indicate a t-ratio large enough in absolute value to reject the null hypothesis of non-stationarity at the 5% level. The critical value (for 100 observations) is given in Fuller (1976) as 2.89. There are only live rejections out of thirty and of those live three fail to reject at the 1% level. If asymptotic normality is invoked, three additional rejections occur.
S.J. Brown et al., The determination Table Estimation
of regional
1 results.
DF test for stationarity of employment
Sector
629
base
DF test for stationarity of location quotient
Average location
Construction
- 1.59
2.19
0.83
Manufacturing Furniture Stone, clay and glass Primary metals Fabricated metals Machinery Electrical Transportation equipment Instruments Food Textiles Apparel Paper Printing and publishing Chemicals Petroleum Miscellaneous
-1.29 - 1.27 -0.46 - 1.33 0.43 - 4.98” - 2.70 - 0.63 -1.15 0.13 0.35 -1.60 - 3.35” -3.18” -1.68 - 1.61
2.12 3.35” 3.87” 3.46” 2.36 3.08” 4.24” 0.97 2.94” 0.66 1.68 1.17” 2.11 2.79 6.47” 6.52”
0.77 0.96 0.99 1.07b 0.99 0.99 0.61 1.36b l.Olb 0.68 l.llb 1.30b 1.40b 1.59b 2.75b 0.58
Trade Wholesale trade Retail trade
- 0.08 - 1.01
4.38” 5.96”
1.06b 0.95
Services Health services Non-health services
- 1.63 0.13
2.96” 7.18”
1.23b l.llb
0.036 1.98
2.89” 4.02”
1.04b l.18b
- 3.52”
10.88”
- 2.36 - 6.66”
4.36” 5.88”
Financial Banking Non-banking Transportation
and public utilities
Government Federal State and local ‘Greater in absolute value than 5% critical ‘Included as a basic sector.
0.91 1.24b 0.92
value of Fuller (1976).
The summary judgement, therefore, is that the raw employment series are generally non-stationary processes, and in particular basic and total employment, being (weighted) sums of these series are also non-stationary, hence the method discussed above has some empirical content. Turning to the problem of defining the base, we use traditional methods to find potential base definitions. We employ four such definitions. BASE1 defines the base as durable manufacturing, and BASE2 defines it as total manufacturing. These definitions are of course quite naive, but as noted above, have been used as potential definitions in some contexts. The other
630
S.J. Brown et al., The determination of regional base
two bases use location quotients as identifiers of more disaggregated sectors. The location quotient for the ith industry, LQ,, is given as
LQi =
Philadelphia
employment
Total Philadelphia
in industry
employment
i U.S. employment
basic
in industry
Total U.S. employment
i ’
The idea is that if LQi is greater than one, Philadelphia is producing more than the ‘expected’ amount of output in that sector, hence the excess must be geared toward export markets. The basic employment in the sector, Bi, can be given by this excess employment [Sullivan (1990)] as
B,=LQi-l
E.
I
LQi
’
(13)
if LQ,> 1. The sum of these Bls then creates the base. Previous usage of location quotients has been confined to data with one observation from the time series. In this context, where particular attention is paid to time series, the location quotient can be computed at each point in time. This brings up the issue of which location quotient to use, and this turns on the question of whether location quotients are themselves stationary or not. If they are, then in the long run they will revert to their unconditional means, which can be estimated from the sample means. This sample mean of LQ can then be appropriately used in the calculation of Bi in (13). If LQi is non-stationary, it becomes unclear what ‘the’ LQi is; for forecasting purposes it will presumably be the last observation since the optimal forecast at any forecast horizon for a random walk process is simply the current realization. Eq. (12) was estimated, with an intercept, for location quotients in each of the twenty-live sectors and the Dickey-Fuller test statistic is given in column 2. A substantial majority, 18 out of 25, exceed 2.89. If asymptotic normality can be invoked, 23 out of 25 would reject nonstationarity. These results indicate that it is indeed appropriate to consider location quotients as stationary processes, and hence we take those industries with average location quotients greater than one as the basic sectors. BASE3 is then calculated using (13) and summing across basic sectors. Tiebout (1962) compared location quotient and survey methods of determining Bi and found that in many cases location quotients severely underestimated the portion of Ei which is export oriented. Many metropolitan economies export nearly all of certain manufacturing sectors’ output, even when (LQ,- l)/LQi is 0.4 or even less. BASE4 attempts to correct for this by taking B as the sum of all sector employment in those sectors previously identified as basic. Each of these four definitions was used as the right-hand variable in a
631
S.J. Brown et al., The determination of regional base Table 2 Regression
results.
Base definition
Cointegrating regression intercept
BASE1 Durable
2,829 (29.41)
- 4.05 ( - 9.48)
- 1.60
2,946 (35.26)
- 2.42 (12.33)
- 1.91
BASE3 Excess LQ method
583 (17.79)
- 3.26
BASE4 Total LQ method
753 (30.0)
9.32 (40.8) 1.14 (3.62)
BASE5 Total LQ; excess services
166 (5.37)
2.43 (57.38)
manufacturing
BASE2 Manufacturing
aGreater
in absolute
value than 5% critical
Parameter estimates slope
Cointegration test statistic
- 3.62” - 4.03”
value of Engle and Yoo (1987).
cointegrating variable with total Philadelphia employment as the regressand. The results are contained in table 2 which gives the slope and intercept (along with t-statistics, though these are biased estimates), and in the third column the Dickey-Fuller test. Rejection (a high statistic) implies a rejection of the hypothesis of no cointegration, which implies ‘acceptance’ of this base definition. The critical value at 5% is given by Engle and Yoo as 3.37. Considering first the naive assignment schemes BASE1 and BASE2, it is clear that these are unsatisfactory base definitions. The cointegration test statistics are well below (in absolute value) the critical value. Furthermore, the indicated multipliers are negative, which is hardly sensible. Rising overall employment coupled with downward drifts in some manufacturing sectors presumably caused this puzzling correlation. The high degree of serial correlation is notable in fig. 1 which plots actual and fitted values of total employment. As noted above the twelve observations of 1987 are reserved for post-sample forecasting. Post-sample fitted values are calculated using actual values of the base in question so the dashed lines should be considered conditional forecasts. Examination of fig. 1 indicates the post-sample performance is quite poor; actual employment appears to be drifting away from both base definitions which is not unexpected given the above results. The location quotients do a rather better job. Table 2 indicates that the excess LQ measure yields a base multiplier of 9.32. Traditional methods of base determination have usually calculated multipliers to be less than four [Isserman (1980)] so that the estimate here seems rather large. The base definition is presumably too small, so that Tiebout’s criticism is presumably justified. The Dickey-Fuller statistic is in fact below the 5% critical value, so that applying the strict test this base definition would not be justified.
S.J. Brown et al., The determination of regional base
Fig.
1
However, it is close, and certainly above the 10% value so caution would seem in order. The total LQ definition, BASE4, does a little better. First, it passes our test, as its Dickey-Fuller statistic is above the critical value, hence the strict test indicates justification for this base. Second, fig. 2 indicates that the within and post-sample performance of this definition is rather superior to all three previous definitions. Third, the base multiplier estimate is reduced substantially to 1.14. Even if the significance of the test is increased to 10% and BASE3 is also deemed ‘appropriate’, the analysis of the previous section may now be applied. It may be the case that BASE3 is too small, with cointegration being attained because some local sectors are cointegrated. Such combinations are straightforward to find. However, with regard to BASE4, three difficulties remain. First, 1.14 is not really a credible multiplier, especially in a metropolitan area the size of Philadelphia. Second, the post-sample forecast of total employment is not particularly good, though it’s better than other definitions employed here. Substantial underprediction remains a problem. Third, BASE4 is not, on closer examination, a completely sensible definition of the base either. This is because it includes all of certain sectors which must, of necessity, be at least in part local serving. This applies most of all to non-health services which,
633
S.J. Brown et al., The determination of regional base 2240
2160
2080 2 E z a 5
2000
1 TOT FIT3 FIT4
-
1
------
I
5; :: L,
1920
I840
I760
1 L
,“‘,~“,~“1~“,“‘,“‘,“‘,“‘,“‘,“‘I”’,”r I975 I977
1
T 1979
1981
I983
I985
I 987
year Fig. 2
because it is a highly aggregated employment series (in itself a problem for proper base identification) contains employment in personal and business services which are of necessity local in character. Though Philadelphia is a central place, and presumably does export these kind of products, some part must be local serving. Hence we create BASE5 as a compromise definition which meets Tiebout’s criticism by including in basic employment all employment in sectors previously identified as basic, but only the ‘excess’ employment for non-health services [as defined in (13)]. The results are quite sensible. First, the multiplier is a reasonable 2.43. The Dickey-Fuller teststatistic increases substantially to 4.03, which is well above the Engle-Yoo critical value, and close the the 1% critical value of 4.07; hence there is every indication that this is a useful base definition. Furthermore, while BASE5 still underpredicts in the post-sample period, the underprediction is not nearly so severe as before, and the prediction errors are about the same size as the in-sample errors (see fig. 3). Finally, it is worth noting that similar further adjustments to other nonmanufacturing, potentially local serving, sectors do not appear to sharpen the definition of the base. These sectors are apparently strongly export oriented. Converting the base contribution of health services, or banking and financial services, for example to ‘excess’ employment reduces the Dickey-Fuller
S.J. Brawn et al., The determination of regional base
634
2220
2160
TOT
~
:,Ts
----
2100
;
2040
E a g
1980
: 9 :: T ‘,
1920
1860
1 BOO
I740
I
” I975
I ”
i
” I977
I
”
/
“, 1979
”
I
”
1981
I
r
1 “I
1983
I
I985
year
Fig. 3
statistic to 3.0 and below, so that these adjustments do not seem to pass the cointegration test. Thus the result for BASE5 appears to give a sharp definition of the economic base for Philadelphia.
5. Conclusions
The idea in Engle and Granger (1987), that long-run equilibrium can be made operational in a time-series regression by defining it as cointegration between variables, is potentially powerful in many areas in economics where the variables defining the equilibrium are not precisely known. We have used it here to define the export base of a metropolitan area, using the fact that according to a standard interpretation of base multiplier theory, basic and total employment are in long-run equilibrium. Finding that naive assignment of manufacturing (alone) to the base does not yield cointegration we use location quotients to pick out those sectors belonging to the base, and find, in accordance with well-established principles from regional economics, that this definition based on location quotient does pass the cointegration test. However, blind application of LQ techniques needs, as always, to be tempered by judgement of the nature of local and basic sectors.
S.J. Brown et al., The determination
of regional base
635
References Brown, Scott J., N. Edward Coulson and Robert F. Engle, 1987, Non-cointegration and econometric evaluation of regional shift and share, Manuscript. Engle, Robert F. and C.W.J. Granger, 1987, Cointegration and error correction: Representation, estimation, and testing, Econometrica 55, 251-276. Engle, Robert F. and B. Sam Yoo, 1987, Forecasting and testing in cointegrated systems, Journal of Econometrics 35, 143-159. Fuller, Wayne, 1976, Introduction to statistical time series (Wiley, New York). Glickman, Norman J., 1977, Econometric analysis of regional systems (Academic Press, New York). Granger, C.W.J. and Paul Newbold, 1974, Spurious regressions in econometrics, Journal of Econometrics 26, 10451066. Hylleberg, Svend and Graham Mizon, 1989, A note on the distribution of the least squares estimation of a random walk with drift, Economics Letters 29, 225-230. Isserman, Andrew, 1980, Estimating export activity in a regional economy: A theoretical and empirical analysis of alternative methods, International Regional Science Review 5, 155-184. Mathur, Vijay K. and Harvey S. Rosen, 1974, Regional employment multipliers: A new approach, Land Economics 50,93-96. Nelson, Charles R. and Charles Plosser, 1982, Trends and random walks in macroeconomic time series, Journal of Monetary Economics 10, 139-162. Nijkamp, Peter, Piet Rietveld and Folke Snickars, 1986, Regional and multiregional economic models: A survey, in: P. Nijkamp, ed., Handbook of regional and urban economics, Vol. 1 (North-Holland, Amsterdam). Richardson, Harry, 1985, Economic base and input-output models, Journal of Regional Science 25, 607-662. Sargan, J.D. and A. Bhargava, 1983, Testing residuals from least squares regression for being generated by a Gaussian random walk, Econometrica 51, 153-174. Sasaki, Kjohei, 1963, Military expenditures and the employment multiplier in Hawaii, Review of Economics and Statistics 45, 298-304. Stock, James H., 1987, Asymptotic properties of least squares estimators of cointegrating vectors, Econometrica 55, 103>1056. Sullivan, Arthur H., 1990, Urban economics (Irwin, Homewood, IL). Tiebout, Charles M., 1962, The community economic base study (Committee for Economic Development, New York). West, Kenneth, 1988, Asymptotic normality, when regressors have a unit root, Econometrica 56, 1397-1418.
Science
and Urban
Economics
22 (1992) 619-635.
North-Holland
On the determination of regional and regional base multipliers*
base
Scott J. Brown Pat@
First Bank, Seattle WA, USA
N. Edward
Coulson
The Pennsylvania
State Unioersity, University Park PA, USA
Robert
F. Engle
University of California at San Diego, La Jolla CA, USA Received
August
1989, final version
received
February
1992
Given that employment in a region’s economic base and total employment are in long-run equilibrium, the two series ought to be cointegrated. This suggests a method for discovering the region’s basic sectors which is explored using the Philadelphia MSA as an example.
1. Introduction
The base multiplier is one of the oldest, simplest, most intuitive and practical models of regional aggregate economic behavior in regional science. For discussions of the merits and faults of this model, there are recent surveys by Richardson (1985) and Nijkamp et al. (1986). Both of these point out that the most difficult and important problem in base analysis is that of defining the base - that is, that portion of regional supply which is devoted to exports. Methods of base identification which have been devised sometimes lack theoretical justification and none seem to be derived from the properties of the multiplier model itself. The method proposed here does not have this defect. In what follows the fact that the base multiplier model is a model of long-run equilibrium is used to yield a stationarity condition on the residual of the regression equation used to estimate the base multiplier. Then basic and total earnings are said to be cointegrated, and so cointegrating tests can be used to test whether the definition of the base is correct. In Correspondence to: ings, The Pennsylvania *We would like to helpful comments, and 016&0462/92/$05.00
Edward Coulson, Department of Economics, 608 Kern State University, University Park, PA 16802, USA. thank Sam Yoo, Mark Watson, and the anonymous Mark Hutchens for research assistance.
0
1992-Elsevier
Science Publishers
Graduate referees
B.V. All rights reserved
Buildfor very
S.J. Brown et al., The determination of regional base
620
section 2, the base multiplier is examined and the test is outlined. In section 3 the problem of base identification is explored and some difficulties of the test examined, followed in section 4 by an empirical example using data from the Philadelphia metropolitan area. Section 5 concludes. 2. The base multiplier model The simple mechanics of the base multiplier model are well known to every regional economist. Some regional aggregate - income, employment and earnings are three examples - is divided into two components: that portion which is exogenously generated by export demand, and that which is endogenously generated by local demand. The first component, which is also called basic demand, provides injections of income into the regional economy, which induces further local spending and a subsequent multiplicative effect in the standard Keynesian fashion. For specificity, and because the empirical example below deals with it, let the aggregate measure of interest be regional employment (E). Then we can write
(1)
E=B+L,
where B is regional is locally generated behavioral equation
employment generated by basic (export) demand and L employment. We can add to this definitional equation a analogous to the Keynesian consumption function:
C=a+bY,
(2)
where in this case Y is total regional income and C is final demand for local goods and services. In this scenario, b is the marginal propensity to consume locally (i.e., neither save nor import from other regions) and a, which is often represents autonomous local demand. suppressed in multiplier studies, Letting 8 be labor’s share of output divided by average wages, multiplying through eq. (2) by this parameter yields L=aO+bE,
(3)
since E = 8Y and L= BC by our definitions. substituting we have the reduced form’ ad E=(l-b)+(l-b)B’
Appropriately
rearranging
and
1
‘Because we assume the region acts as a small open economy earnings can be assumed to be exogenous, though this is not critical below.
within its nation, export for the cointegration tests
S.J. Brown et al., The determination of regional base
621
so that l/( 1 -b) is the (marginal) base multiplier which gives the increase in total regional employment given a unit increase in B, and is the parameter of interest here. This model is quite simple, and it can be extended in any number of ways. Nevertheless, it is a good prototype, in particular since it yields a constant marginal multiplier which seems a reasonable approximation and is often used in practical applications by regional analysts. The practicality of the base multiplier model is its most obvious attribute [see Nijkamp et al. (1986)]. The question then is, how is one to estimate the base multiplier for a specific region? One obvious first step is to treat (4) as a regression equation, i.e., rewrite it (adding an error term) as E=cr+flB+o,
(5)
and then gather (for the particular region) a time series of observations on E and B, run the regression and obtain estimates of the multiplier, p. However, this leaves aside two questions of major importance: (a) How is B calculated? and (b) What will the properties of the error term be? With regard to (a), which, as noted, is the major issue in the application of the base multiplier model, the ultimate, though very expensive answer, would be to sample a number of firms, presumably stratified by Standard Industrial Classification (SIC) code, and apply the proportion of sales the sampled firms give as export-oriented to the aggregate employment of that sector, then adding up the export employment from the various sectors to obtain B. However, this will obviously be very costly, and likely to be infeasible for many researchers. Richardson (1985) examines several non-survey methods of obtaining estimates of B. The simplest and crudest of these is the assignment method, wherein the earnings within broad employment categories are wholly assigned to either basic or local earnings. Often, manufacturing employment is assigned to the base and all other employment is assumed to be locally generated. While this seems particularly crude, it should be noted that this assumption underlies the structural econometric models of Glickman (1977) insofar as such models contain equations where manufacturing output (and only manufacturing output) is solely a function of macroeconomic variables. Other methods of base determination do not make the assumption that an industry’s employment must be entirely basic or local, but rather use some secondary source of information to determine the proportion of employment to be assigned to each. The secondary source is often national employment in the industry, which can be used to calculate location quotients.’ However users of either the assignment method or any of the suggested *See also Mathur
and Rosen (1974).
622
S.J. Brown et al., The determination of regional base
methods of division have no way of checking the adequacy of their base measurement, so that (as has been noted) estimation of the base multiplier is likely to be highly dependent on what are, of necessity, ad hoc assumptions about the nature of the base. This brings us to the second of the two difficulties mentioned above. Supposing that the problem of base identification has been solved, what are the consequences of estimating (5) in a time-series framework? Such a regression is indeed fraught with difficulties, the major one being that it is quite probable that both E and B are non-stationary series - that is, they exhibit stochastic trends. Certainly national aggregates follow such trends as the results of Nelson and Plosser (1982) indicate. Therefore at least some of the component regional aggregates must do so as well. The difficulties of estimation and inference in regression equations such as (5) when one nonstationary series is regressed on another have been well-documented in Granger and Newbold (1974) who call such regressions ‘spurious’. They note that two series might be independent random walks (perhaps with drift) having no other relationship whatsoever, and yet when one is regressed on another, the resulting R2 and t-statistic can be extremely high, and lead the investigator to conclude that a ‘good lit’ has been obtained. The danger for base multiplier estimation is clear: Any base calculation, correctly or incorrectly done, will appear to lit well with the data and perhaps lead one to assume that the true base has indeed been found, and that the base multiplier has been consistently and efficiently estimated. This danger is detectable because the properties of the residual series in spurious regressions will certainly not conform to the ideal conditions. In particular, since linear combinations of non-stationary series are typically non-stationary themselves, the particular linear combination E, - &- fill, = il, will have this property. Thus the residuals will be non-stationary and so have infinite variance and undefined mean (in population) which hardly lends itself to accurate inference. On another level, if the residual of the regression has an infinite variance, then it would seem that E and B (or rather a linear function of B) can drift infinitely far apart, which would suggest that B and E in fact do not have an (long-run) equilibrium, which is certainly a contradiction of the idea underlying the base multiplier model. However Engle and Granger (1987) have pointed out that it need not be the case that linear combinations of non-stationary variables are themselves non-stationary. Certain combinations may be such that the non-stationary parts ‘cancel each other out’ leaving only stationary residuals. If such a linear combination exists, the two variables are said to be cointegrated and the existence of this possibility has crucial ramifications for our problem. For consider that if 6,, the estimated residuals, are stationary, then they have finite variance, with the attendant implication that E, and B, will drift infinitely far apart only with infinitely small probability. Engle and Granger
S.J. Brown et al., The determination of regional base
623
(1987) note that this provides an operational definition for (long-run) equilibrium in an econometric context, for if two or more variables are in long-run equilibrium, then even if there are continual random shocks to the equilibrium system the variables should move together over time. This will be the case if u, is stationary, and will not if U, is non-stationary. With the presumption that the base multiplier model is indeed a model of long-run equilibrium between E and B, the above discussion contains a potentially powerful suggestion for the problem of base determination. B can be necessarily defined as the set of earnings flows which cointegrates with E in estimation of (5). Thus we can undertake a search process, estimating eq. (5) under OLS using several different definitions of the base, and test the residuals for stationarity.3 If the residuals are non-stationary, then E and B do not cointegrate, they can presumably drift infinitely far apart, and they do not form the long-run equilibrium. Thus the given definition of B cannot be the true definition of the base. By contrast, stationarity in the residuals would indicate that the true B has been found. Notice also that a consistent estimate of the multiplier itself has also been found in the estimate of (5). In fact Stock (1987) suggests that this estimate is ‘super-consistent’ since the estimator converges to a faster rate than standard OLS estimators. However it turns out that the t-statistics may be far from the standard distributions, particularly in small samples. The short-run dynamics should be modeled, presumably in an error correction framework as suggested by Engle and Granger (1987). Special cases of the error correction mechanism, such as the partial adjustment model, have been previously used in this context [Sasaki (1964)].
3. Cointegration
and problem of base identification
In this section we wish to expand the discussion of the preceding section. More particularly, the connection between stationarity of the residuals in the cointegrating regression (5) and the identification of the base is subject to two difficulties, which correspond to the standard type I and type II errors: (1) Is it possible to reject a base definition when it is correct? and (2) Is it possible to accept a base definition when it is wrong? As we shall see in the next section where we discuss estimation issues, the test for the stationarity of the residuals has as its null hypothesis the existence of a unit root in the lag polynomial which models the residual
‘Such a search process raises the question of be, which we ignore in our empirical example. correlated (even if it is stationary) so that the only in the long run. Thus we can only describe
what the nominal sizes of successive tests should It should be noted that v, will be highly serially variables will return to the equilibrium position E = a + /3B as a long-run equilibrium.
S.J. Brown et al., The determination of regional base
624
series. Thus the null hypothesis is one of non-stationarity. While this is a non-standard testing problem, Engle and Granger (1987) and others have developed tests, the Monte Carlo simulations of which generate appropriate critical values for tests of standard sizes. In attempting to answer (l), the appropriate question is whether it is possible to accept the null hypothesis of non-stationarity when in fact the residuals are stationary. Thus the power of the tests become important. While tests for non-stationary residuals are well known to have low power for close alternatives, the simulation results of Engle and Granger show that some of the tests have reasonable power against alternatives which are not too close. The importance of power against close alternatives depends on the importance placed on the distinction between an equilibrium which takes a very long time to achieve, and a dynamic model which has no equilibrium. In practice this distinction may not be observable in a finite amount of data [Nelson and Plosser (1982)], and may not be important. While the issues surrounding question (1) are fairly clearcut (even if the answer is not) the analysis of question (2) is somewhat more subtle. The idea is that while a correct base definition is sufficient to yield a stationary residual, it is not necessary if there are other cointegrating relationships among the variables in the base multiplier model. Since arbitrary finite linear combinations of stationary variables will yield stationary variables in turn, if the base multiplier model (5) replicates a linear combination of other cointegrating equations, then a stationary error would result, even if the base definition used is incorrect. To see this, consider a simple region with four industries, with employment xi,i=l,..., 4. Let X, and X, be basic sectors so that B= X, +XZ and
(1 -P)W, +X,)+X,+X,=0,, where v, is a stationary sequence.4 The above equation is merely a reformulation of (5) for this simple scenario. If the investigator mistakenly identifies the base as X, +X,+X, and (6) is the only linear combination of the earnings variables which yields a stationary residual, then by forcing X, to have a coefficient of (1 -p) rather than one (noting /3#0) the investigator must automatically observe a non-stationary residual and would thereupon recognize that the base has not been correctly identified. However, (6) may not be the sole cointegrating relationship. For example, the two local sectors may be growing together over time; that is x4--Y,x,=o,:
41n what follows the intercept estimated regressions below.
term is suppressed
for simplicity,
though
it is included
in the
625
S.J. Brown et al., The determination of regional base
may be a cointegrating incorrect base multiplier
equation mode1
with
(l-p*)(x,+x~+x,)+x,=w, is estimated. OLS will search for that and so will search for cointegrating yield infinite variances (in population). In this case the answer is yes. Let zi and (7) and solve the following set of a free parameter
v2 stationary.5
Now
suppose
the
(8) fl* which minimizes the variance of w,,, vectors in preference to vectors which Does such a linear combination exist? and z2 be the weights given to eqs. (6) linear equations, noting that /I* is also
z,(l -P)=(l-P*)>
(94
zt(l -P)=(l -P*),
(9’4
z,-Yz2=(1-B*),
(9c)
z,+z,=l.
(9d)
Since (9a) and (9b) are dependent equations we are left with three equations and three unknowns which have the solution zr = y//3+ y, z2 =/?//I+ y and /I* =( 1+7)/I/(/I + y). Thus, if (8) is estimated instead of (6), the OLS estimation method will pick off the given /I* because this will yield the stationary residual we = (y/j? + y)v, +(/I/j? + y)v,_ Even though a stationary residual has been estimated, the base has not been correctly identified, and noting that /I > 1, /I* < fi. That is, the multiplier has been underestimated, which of course is a standard result when too much has been included in the base. Consider, on the other hand, the possibility of including too little in the base. Suppose the investigator estimates
x,(1-~*)+x,+x,+x,=w”,
(10)
in which case we have
z,(l -B)=(l-8*), z,(l-B)=l, z1
-yzz
= 1,
*The suggestion is based on the presumption that supply and demand for the region’s locally produced services are not subject to permanent taste or productivity shocks so that the ratio of earnings stays (approximately) constant over time.
S.J. Brown et al., The determination of regional base
626
Zl
+z, = 1,
and it is easy to see that (even aside from the fact that we have four equations and three unknowns) unless fi* =y =O, these equations have no solution. However, we are being somewhat too restrictive here. We are allowed to postulate up to three cointegrating relationships since we have four variables in the system: Thus along with allowing the two local sectors to grow together over time, let us allow the two basic sectors to do so as we11.6 We add
(11)
x1-ljx2=ug to the system of structural equations, with uj stationary. a staionary residual residual if there is a solution to
z,(l -P)-z,= z1=
1,
z1=
1.
Then (10) will have
1,
(Note that z2 must equal zero here.) The solution is clearly zr = 1, zs =/3 yielding p* =fi( 1+$), and since $ >O, /I* >/I, the standard result when too little is included in the base. Examples with greater numbers of basic and local sectors can also be constructed; the analysis becomes more complex because a richer variety of cointegrating relationships can be analyzed. The general conclusions to be derived from such analysis are: (1) false acceptance of too small a base is impossible when basic sectors are not cointegrated; (2) false acceptance of too large a base is impossible when local sectors are not cointegrated; (3) false acceptance of a base is more likely, the larger the rank of the cointegrating matrix, i.e., the larger the number of cointegrating relationships. (1) and (2) can be seen from expanding the above four-industry analysis to an arbitrarily larger number. (3) arises from the fact that when there are fewer cointegrating relationships among (say) the basic sectors, it is obviously more difficult to replicate a linear combination of them in any given cointegrating regression. 6As in footnote 5, the possibility that the two export sectors are cointegrated is based on the absence of permanent taste or productivity shocks. If it is the case that such shocks are indeed present so that secondary cointegrating relationships do not exist, then (6) is the only cointegrating equation and the stationarity of the residual is suflicient to define the base.
S.J. Brown et al., The determination of regional base
627
We now address the question of the division of sector earnings into its basic and local components. Letting pi be the portion of the ith sector devoted to exports, the base multiplier model can be written as
but obviously the model as it stands is not identified. A least squares projection would simply give back $J= 1 for all i. Some prior information on the 4i as derived, for example, from location quotients, is necessary. The advantage of the cointegration technique described here is that if the prior information is wrong, the regression will return to a non-stationary error, and the researcher will know that the true base definition has not been found. Secondly, there is the issue of allowing differential multipliers across industries. Here the model would be
and the identification problem is obviously more severe. The amount of prior information necessary depends on the variability of the pi. In the extreme case where each sector’s multiplier is separately estimated, each 4i must be calculated beforehand; we then have the reduced form of the standard inputoutput model. 4. Estimation The specification of the base in a base multiplier model must necessarily proceed in three steps. First, it must be ascertained that the employment series under consideration (total employment and whatever passes for basic employment) are indeed non-stationary processes. Second, the various detinitions of export base must be calculated. Third, cointegration tests must be performed, to see if the residuals are stationary, as required by multiplier theory. The tests of stationarity required in the first and third parts (and as it turns out, the second as well) are generally assumed to be first order tests, that is, whether the variable of interest is a random walk (perhaps with drift) or not. Therefore, in the language of classical statistics, the test is of H,: p = 1 in the regression 4t= P4t - 1+ 4,
for some variable qt. If He is rejected in favor of the alternative p < 1, then qt is a stationary process. If not, the random walk null is not rejected. Of the several testing procedures, the one which has found the most favor in the
628
S.J. Brown et al., The determination of regional base
literature is the Dickey-Fuller regression to
test, which involves
the transformation
of the
(12) so that the null is transformed into a zero restriction on the coefftcient of the lagged dependent variable. Then the ordinary t-ratio given in regression output can be used as a test statistic. In the case here, this t-ratio is not distributed as a t, nor is it asymptotically normal, but Fuller (1976) gives critical values based on Monte Carlo simulation. The case above describes the test when the null does not allow drift, and the alternative is a zero-mean stationary process. This is appropriate when q1 is a residual, as would be the case in the cointegration tests. In fact, the cointegrating case is complicated by the fact that the distribution of the Dickey-Fuller test statistic widens as the number of variables in the cointegrating equation increases above one. However Engle and Yoo (1987) provide appropriate critical values for the cointegration case. Testing for the non-stationarity of a raw employment series is complicated by the fact that such series may have drift. Hence an intercept term may be added to (12) above. With a non-zero intercept under the alternative, the series is seen to have a non-zero long-run equilibrium. West (1988) has shown that the slope coefficient is asymptotically normal in this case, even under the null. On the other hand, the simulated Dickey-Fuller critical values for this case appear to be far from standard. Investigations by Hylleberg and Mizon (1989) indicate that the normal distribution is not a good approximation in finite samples unless the drift is large (like 100 times the variance of u,). This does not appear to be the case. We will refer to both normal and Dickey-Fuller results below. Before turning to the estimation, a word about the data is in order. The sample consists of total and sectoral employment at monthly intervals for the Philadelphia Metropolitan Statistical Area from January 197551987. The actual sample period within which the multiplier is estimated is cut off at December 1986 in order to facilitate post-sample comparisons. The sectoral employment series are generated from one and two-digit SIC employment data from Employment and Earnings: States and Areas. The sectors considered are given in column 1 of table 1. In column 2 the Dickey-Fuller test statistic derived from estimation of (12) and including a drift term is given for each of these sectors’ raw employment series. Asterisks there indicate a t-ratio large enough in absolute value to reject the null hypothesis of non-stationarity at the 5% level. The critical value (for 100 observations) is given in Fuller (1976) as 2.89. There are only live rejections out of thirty and of those live three fail to reject at the 1% level. If asymptotic normality is invoked, three additional rejections occur.
S.J. Brown et al., The determination Table Estimation
of regional
1 results.
DF test for stationarity of employment
Sector
629
base
DF test for stationarity of location quotient
Average location
Construction
- 1.59
2.19
0.83
Manufacturing Furniture Stone, clay and glass Primary metals Fabricated metals Machinery Electrical Transportation equipment Instruments Food Textiles Apparel Paper Printing and publishing Chemicals Petroleum Miscellaneous
-1.29 - 1.27 -0.46 - 1.33 0.43 - 4.98” - 2.70 - 0.63 -1.15 0.13 0.35 -1.60 - 3.35” -3.18” -1.68 - 1.61
2.12 3.35” 3.87” 3.46” 2.36 3.08” 4.24” 0.97 2.94” 0.66 1.68 1.17” 2.11 2.79 6.47” 6.52”
0.77 0.96 0.99 1.07b 0.99 0.99 0.61 1.36b l.Olb 0.68 l.llb 1.30b 1.40b 1.59b 2.75b 0.58
Trade Wholesale trade Retail trade
- 0.08 - 1.01
4.38” 5.96”
1.06b 0.95
Services Health services Non-health services
- 1.63 0.13
2.96” 7.18”
1.23b l.llb
0.036 1.98
2.89” 4.02”
1.04b l.18b
- 3.52”
10.88”
- 2.36 - 6.66”
4.36” 5.88”
Financial Banking Non-banking Transportation
and public utilities
Government Federal State and local ‘Greater in absolute value than 5% critical ‘Included as a basic sector.
0.91 1.24b 0.92
value of Fuller (1976).
The summary judgement, therefore, is that the raw employment series are generally non-stationary processes, and in particular basic and total employment, being (weighted) sums of these series are also non-stationary, hence the method discussed above has some empirical content. Turning to the problem of defining the base, we use traditional methods to find potential base definitions. We employ four such definitions. BASE1 defines the base as durable manufacturing, and BASE2 defines it as total manufacturing. These definitions are of course quite naive, but as noted above, have been used as potential definitions in some contexts. The other
630
S.J. Brown et al., The determination of regional base
two bases use location quotients as identifiers of more disaggregated sectors. The location quotient for the ith industry, LQ,, is given as
LQi =
Philadelphia
employment
Total Philadelphia
in industry
employment
i U.S. employment
basic
in industry
Total U.S. employment
i ’
The idea is that if LQi is greater than one, Philadelphia is producing more than the ‘expected’ amount of output in that sector, hence the excess must be geared toward export markets. The basic employment in the sector, Bi, can be given by this excess employment [Sullivan (1990)] as
B,=LQi-l
E.
I
LQi
’
(13)
if LQ,> 1. The sum of these Bls then creates the base. Previous usage of location quotients has been confined to data with one observation from the time series. In this context, where particular attention is paid to time series, the location quotient can be computed at each point in time. This brings up the issue of which location quotient to use, and this turns on the question of whether location quotients are themselves stationary or not. If they are, then in the long run they will revert to their unconditional means, which can be estimated from the sample means. This sample mean of LQ can then be appropriately used in the calculation of Bi in (13). If LQi is non-stationary, it becomes unclear what ‘the’ LQi is; for forecasting purposes it will presumably be the last observation since the optimal forecast at any forecast horizon for a random walk process is simply the current realization. Eq. (12) was estimated, with an intercept, for location quotients in each of the twenty-live sectors and the Dickey-Fuller test statistic is given in column 2. A substantial majority, 18 out of 25, exceed 2.89. If asymptotic normality can be invoked, 23 out of 25 would reject nonstationarity. These results indicate that it is indeed appropriate to consider location quotients as stationary processes, and hence we take those industries with average location quotients greater than one as the basic sectors. BASE3 is then calculated using (13) and summing across basic sectors. Tiebout (1962) compared location quotient and survey methods of determining Bi and found that in many cases location quotients severely underestimated the portion of Ei which is export oriented. Many metropolitan economies export nearly all of certain manufacturing sectors’ output, even when (LQ,- l)/LQi is 0.4 or even less. BASE4 attempts to correct for this by taking B as the sum of all sector employment in those sectors previously identified as basic. Each of these four definitions was used as the right-hand variable in a
631
S.J. Brown et al., The determination of regional base Table 2 Regression
results.
Base definition
Cointegrating regression intercept
BASE1 Durable
2,829 (29.41)
- 4.05 ( - 9.48)
- 1.60
2,946 (35.26)
- 2.42 (12.33)
- 1.91
BASE3 Excess LQ method
583 (17.79)
- 3.26
BASE4 Total LQ method
753 (30.0)
9.32 (40.8) 1.14 (3.62)
BASE5 Total LQ; excess services
166 (5.37)
2.43 (57.38)
manufacturing
BASE2 Manufacturing
aGreater
in absolute
value than 5% critical
Parameter estimates slope
Cointegration test statistic
- 3.62” - 4.03”
value of Engle and Yoo (1987).
cointegrating variable with total Philadelphia employment as the regressand. The results are contained in table 2 which gives the slope and intercept (along with t-statistics, though these are biased estimates), and in the third column the Dickey-Fuller test. Rejection (a high statistic) implies a rejection of the hypothesis of no cointegration, which implies ‘acceptance’ of this base definition. The critical value at 5% is given by Engle and Yoo as 3.37. Considering first the naive assignment schemes BASE1 and BASE2, it is clear that these are unsatisfactory base definitions. The cointegration test statistics are well below (in absolute value) the critical value. Furthermore, the indicated multipliers are negative, which is hardly sensible. Rising overall employment coupled with downward drifts in some manufacturing sectors presumably caused this puzzling correlation. The high degree of serial correlation is notable in fig. 1 which plots actual and fitted values of total employment. As noted above the twelve observations of 1987 are reserved for post-sample forecasting. Post-sample fitted values are calculated using actual values of the base in question so the dashed lines should be considered conditional forecasts. Examination of fig. 1 indicates the post-sample performance is quite poor; actual employment appears to be drifting away from both base definitions which is not unexpected given the above results. The location quotients do a rather better job. Table 2 indicates that the excess LQ measure yields a base multiplier of 9.32. Traditional methods of base determination have usually calculated multipliers to be less than four [Isserman (1980)] so that the estimate here seems rather large. The base definition is presumably too small, so that Tiebout’s criticism is presumably justified. The Dickey-Fuller statistic is in fact below the 5% critical value, so that applying the strict test this base definition would not be justified.
S.J. Brown et al., The determination of regional base
Fig.
1
However, it is close, and certainly above the 10% value so caution would seem in order. The total LQ definition, BASE4, does a little better. First, it passes our test, as its Dickey-Fuller statistic is above the critical value, hence the strict test indicates justification for this base. Second, fig. 2 indicates that the within and post-sample performance of this definition is rather superior to all three previous definitions. Third, the base multiplier estimate is reduced substantially to 1.14. Even if the significance of the test is increased to 10% and BASE3 is also deemed ‘appropriate’, the analysis of the previous section may now be applied. It may be the case that BASE3 is too small, with cointegration being attained because some local sectors are cointegrated. Such combinations are straightforward to find. However, with regard to BASE4, three difficulties remain. First, 1.14 is not really a credible multiplier, especially in a metropolitan area the size of Philadelphia. Second, the post-sample forecast of total employment is not particularly good, though it’s better than other definitions employed here. Substantial underprediction remains a problem. Third, BASE4 is not, on closer examination, a completely sensible definition of the base either. This is because it includes all of certain sectors which must, of necessity, be at least in part local serving. This applies most of all to non-health services which,
633
S.J. Brown et al., The determination of regional base 2240
2160
2080 2 E z a 5
2000
1 TOT FIT3 FIT4
-
1
------
I
5; :: L,
1920
I840
I760
1 L
,“‘,~“,~“1~“,“‘,“‘,“‘,“‘,“‘,“‘I”’,”r I975 I977
1
T 1979
1981
I983
I985
I 987
year Fig. 2
because it is a highly aggregated employment series (in itself a problem for proper base identification) contains employment in personal and business services which are of necessity local in character. Though Philadelphia is a central place, and presumably does export these kind of products, some part must be local serving. Hence we create BASE5 as a compromise definition which meets Tiebout’s criticism by including in basic employment all employment in sectors previously identified as basic, but only the ‘excess’ employment for non-health services [as defined in (13)]. The results are quite sensible. First, the multiplier is a reasonable 2.43. The Dickey-Fuller teststatistic increases substantially to 4.03, which is well above the Engle-Yoo critical value, and close the the 1% critical value of 4.07; hence there is every indication that this is a useful base definition. Furthermore, while BASE5 still underpredicts in the post-sample period, the underprediction is not nearly so severe as before, and the prediction errors are about the same size as the in-sample errors (see fig. 3). Finally, it is worth noting that similar further adjustments to other nonmanufacturing, potentially local serving, sectors do not appear to sharpen the definition of the base. These sectors are apparently strongly export oriented. Converting the base contribution of health services, or banking and financial services, for example to ‘excess’ employment reduces the Dickey-Fuller
S.J. Brawn et al., The determination of regional base
634
2220
2160
TOT
~
:,Ts
----
2100
;
2040
E a g
1980
: 9 :: T ‘,
1920
1860
1 BOO
I740
I
” I975
I ”
i
” I977
I
”
/
“, 1979
”
I
”
1981
I
r
1 “I
1983
I
I985
year
Fig. 3
statistic to 3.0 and below, so that these adjustments do not seem to pass the cointegration test. Thus the result for BASE5 appears to give a sharp definition of the economic base for Philadelphia.
5. Conclusions
The idea in Engle and Granger (1987), that long-run equilibrium can be made operational in a time-series regression by defining it as cointegration between variables, is potentially powerful in many areas in economics where the variables defining the equilibrium are not precisely known. We have used it here to define the export base of a metropolitan area, using the fact that according to a standard interpretation of base multiplier theory, basic and total employment are in long-run equilibrium. Finding that naive assignment of manufacturing (alone) to the base does not yield cointegration we use location quotients to pick out those sectors belonging to the base, and find, in accordance with well-established principles from regional economics, that this definition based on location quotient does pass the cointegration test. However, blind application of LQ techniques needs, as always, to be tempered by judgement of the nature of local and basic sectors.
S.J. Brown et al., The determination
of regional base
635
References Brown, Scott J., N. Edward Coulson and Robert F. Engle, 1987, Non-cointegration and econometric evaluation of regional shift and share, Manuscript. Engle, Robert F. and C.W.J. Granger, 1987, Cointegration and error correction: Representation, estimation, and testing, Econometrica 55, 251-276. Engle, Robert F. and B. Sam Yoo, 1987, Forecasting and testing in cointegrated systems, Journal of Econometrics 35, 143-159. Fuller, Wayne, 1976, Introduction to statistical time series (Wiley, New York). Glickman, Norman J., 1977, Econometric analysis of regional systems (Academic Press, New York). Granger, C.W.J. and Paul Newbold, 1974, Spurious regressions in econometrics, Journal of Econometrics 26, 10451066. Hylleberg, Svend and Graham Mizon, 1989, A note on the distribution of the least squares estimation of a random walk with drift, Economics Letters 29, 225-230. Isserman, Andrew, 1980, Estimating export activity in a regional economy: A theoretical and empirical analysis of alternative methods, International Regional Science Review 5, 155-184. Mathur, Vijay K. and Harvey S. Rosen, 1974, Regional employment multipliers: A new approach, Land Economics 50,93-96. Nelson, Charles R. and Charles Plosser, 1982, Trends and random walks in macroeconomic time series, Journal of Monetary Economics 10, 139-162. Nijkamp, Peter, Piet Rietveld and Folke Snickars, 1986, Regional and multiregional economic models: A survey, in: P. Nijkamp, ed., Handbook of regional and urban economics, Vol. 1 (North-Holland, Amsterdam). Richardson, Harry, 1985, Economic base and input-output models, Journal of Regional Science 25, 607-662. Sargan, J.D. and A. Bhargava, 1983, Testing residuals from least squares regression for being generated by a Gaussian random walk, Econometrica 51, 153-174. Sasaki, Kjohei, 1963, Military expenditures and the employment multiplier in Hawaii, Review of Economics and Statistics 45, 298-304. Stock, James H., 1987, Asymptotic properties of least squares estimators of cointegrating vectors, Econometrica 55, 103>1056. Sullivan, Arthur H., 1990, Urban economics (Irwin, Homewood, IL). Tiebout, Charles M., 1962, The community economic base study (Committee for Economic Development, New York). West, Kenneth, 1988, Asymptotic normality, when regressors have a unit root, Econometrica 56, 1397-1418.