The exchange rate and consumer prices in Pakistan: is rupee devaluation inflationary?

Choudhri, Ehsan U. ; Khan, Mohsin S.

This paper challenges the popular view that devaluation of the rupee is inflationary. Recent developments in the theoretical literature are reviewed to explain why consumer prices would be unresponsive to exchange rate changes in the short run. Then empirical tests are conducted for Pakistan during the period 1982 to 2001 to examine whether inflation is systematically related to changes in the exchange rate. The empirical analysis finds no association between rupee devaluations and inflation in Pakistan. It appears, therefore, that concerns about the inflationary consequences of rupee devaluation are unsupported by the facts.

1. INTRODUCTION

Does devaluation lead to an increase in prices? This is a critical question that policy-makers in Pakistan have faced continuously over the past three decades or so, and particularly since 1982, following the adoption of a flexible exchange rate policy. At the beginning of 1972, the US dollar exchanged for about five Pakistani rupees. After a devaluation in 1972 and a small revaluation in 1973, the exchange rate remained fixed at about ten rupees per dollar till the end of 1981. The exchange rate was allowed to vary since January 1982 and it rose to a rate around sixty rupees per dollar over the next two decades. This increase involved a number of sizable rupee devaluations. Such devaluations receive considerable attention and often raise the concern that they would contribute to inflation. The concern about inflation is based on the popular view, which has sometimes been shared by officials in policymaking circles, that consumer prices are significantly affected by imported goods prices, which increase quickly in response to a devaluation. (1) This view is generally thought to find support in empirical studies of inflation in Pakistan. In the most recent study on the subject, Ahmad and Ali (1999) assert that the "recent empirical work in Pakistan provides consistent evidence that the domestic price level responds significantly but gradually to exchange rate devaluation" (p. 237).

Obviously, if the devaluation-inflation link exists, then devaluation comes with an important cost that necessarily must be factored into the exchange rate policy. Furthermore, it implies that the authorities can only affect the real exchange rate temporarily, because as domestic prices rise, the initial effects of a nominal depreciation on the real exchange rate would be reversed. On these two counts at least, exchange rate policy becomes fairly constrained.

This paper argues that the fear of inflation associated with devaluations in Pakistan is largely unfounded. We draw on recent developments in the literature on the exchange rate pass-through and the purchasing power parity to suggest reasons why consumer prices might be unresponsive to changes in the exchange rate. We also reexamine the evidence for Pakistan and present new results, which demonstrate that rupee devaluations have had little impact on inflation. (2) The theory underlying the relationship between the exchange rate and consumer prices is reviewed in Section 2. The empirical analysis is discussed in Section 3. The conclusions of the paper are contained in Section 4.

2. THE EXCHANGE RATE PASS-THROUGH TO CONSUMER PRICES

This section briefly discusses the channels through which changes in the exchange rate pass-through to consumer prices. We begin with the conventional model that implies a significant and, under certain conditions, a complete pass-through to consumer prices in the short run. (3) We then discuss certain departures from the conventional model suggested by recent literature, which could account for little or no pass-through in the short run. Finally, we examine the long-run relationship between the exchange rate and the price level, and discuss conditions that would weaken this relationship.

2.1 Traded and Non-traded Goods

The basket of goods entering the consumer price index (CPI) can be divided into traded and non-traded goods. Letting [P.sub.t] denote the CPI in period t, we can express it as

[P.sub.t] = [([P.sup.T.sub.t]).sup.[theta]] [([P.sup.NT.sub.t]).sup.1-[theta]], ... ... ... ... ...(1)

where [P.sup.T.sub.t] and [[P.sup.NT.sub.t] are the price indexes for traded and non-traded goods, and 0 is the weight assigned to traded goods. The home price of traded goods can be directly linked to the price of foreign traded goods adjusted for the exchange rate. The conventional model assumes that traded goods are produced under competitive conditions. This assumption implies that, in the absence of trade barriers, the price of a traded good expressed in a particular currency will be the same in each country. (4) The following relation can then be derived by aggregating over prices of individual traded goods:

[[P.sup.T.sub.t] = [S.sub.t][[P.sup.T*.sub.t], ... ... ... ... ... ... (2)

where [S.sub.t] is the exchange rate (expressed as the price of foreign currency) and [[P.sup.T*.sub.t] is the foreign price index for traded goods (which is assumed to use the same weights as the home index). This relation implies that a change in the exchange rate will fully pass-through to the traded goods price index. The relation can be easily modified to introduce trade costs (e.g., transportation costs, tariffs, and non-tariff barriers). These costs introduce a wedge between the home and (exchange-rate adjusted) foreign prices, but as long as these costs are determined independently of the exchange rate, the exchange rate pass-through to traded goods prices would continue to be complete.

The price for the non-traded goods sector will be determined by the demand and supply functions for this sector. These functions would depend on the ratio of non-traded goods to traded goods prices. Let [gamma] be the price ratio that clears the nontraded goods market. Thus, in equilibrium

[[P.sup.NT.sub.t] = [gamma][[P.sup.T.sub.t]. ... ... ... ... ... ... (3)

Relation (3) provides an indirect link between the exchange rate and the non-traded goods price index. If the price of non-traded goods is flexible and adjusts quickly, (3) will hold in the short run. In this case, there will be a complete pass-through in the short run to both traded and non-traded goods prices [via (2) and (3)] and hence to the CPI. In fact, assuming that relations analogous to (1) and (3) hold for the foreign economy, we can also link the domestic CPI to both the exchange rate and the foreign CPI. Using (1), (3), their foreign counterparts, and (2), we can obtain

[[P.sub.t] = [[P.sub.t][[P.sup.*.sub.t] [([gamma]/[[gamma].sup.*]).sup.1-[theta]] ... ... ... ... ... ... (4)

where [[P.sup.*.sub.t] and [[gamma].sup.*] represent the CPI and the equilibrium relative price of non-traded goods in the foreign country. The home CPI responds fully to the foreign CPI as well as the exchange rate in this relation.

The short-run impact of the exchange rate on CPI would be weakened if nontraded goods prices are sticky. Exchange rate changes would now affect only traded goods prices in the short run and the short-run pass-through to CPI would equal the share of traded goods, [theta]. (5) The share of traded goods can be substantial and thus the short-run pass-through to CPI can be sizeable even if non-traded goods prices adjust slowly.

2.2 The Short-run Pass-through

We next discuss a number of variations of the above model that could cause the short-run pass-through to be small. Consider, first, the variations that loosen the traded goods price relation (2) in the short run. One important point of departure in the recent literature has been to relax the assumption of perfect competition. Under imperfect competition, prices include a markup over costs and producers have the discretion to vary the markup across countries (i.e., they can price to market). (6) The pass-through to traded goods price would be incomplete if variations in the markup offset changes in the exchange rate. (7) However, the equilibrium markup need not respond systematically to exchange rate changes. In fact, it can be shown that the equilibrium markup would be invariant with respect to the exchange rate under the standard assumption of a constant price elasticity of demand.

Another argument is that imperfectly-competitive producers would change prices infrequently in the presence of even small menu costs. Infrequent price adjustment would not prevent complete pass-through in the importing country if the producer (exporter) fixes the price in its own currency. However, if the price is set in local (importer's) currency and is sticky, it would be unresponsive to exchange rate fluctuations in the short run. (8) Local currency pricing can thus play an important role in blocking the impact of exchange rate changes on traded goods prices in the short run. However, while local currency pricing has been observed in some large industrial countries, such as the United States, it is not clear whether this practice occurs in developing countries like Pakistan.

A weak response of the traded goods component of the CPI may also be explained by the fact that imported goods are generally not sold directly to consumers. Many imported goods are, in fact, intermediate inputs imported by firms producing final products. (9) If prices of the final products are adjusted infrequently, changes in the cost of inputs resulting from fluctuations in the exchange rate will not be quickly passed on to consumers. Even non-intermediate imports go through distribution channels (transportation, marketing, retailing) before they are delivered to consumers. These channels largely use non-traded services, which can account for a large fraction of the consumer price. (10) The price component represented by local services would then not be affected by the exchange rate in the short run.

The short-run pass-through could also be weakened by certain factors that introduce biases in measured prices and are of special relevance to developing economies. For example, a home-currency devaluation could induce a substitution of cheaper lower-quality local goods for imported goods. Burstein, Eichenbaum, and Rebelo (2001) refer to this phenomenon as "flight from quality" and argue that it helps explain why a number of large devaluations had a small impact on the measured inflation rate. Price regulation of "essential" commodities and foreign exchange controls could represent another important source of bias in the observed price and exchange rate data. Because measured values would not fully reflect true market values under these policies, the pass-through relation would be distorted.

2.3 The Long-run Relation

The reasons discussed above can insulate traded goods prices from changes in the exchange rate in the short run. However, they do not explain why traded and non-traded goods prices would not fully respond to exchange rate changes in the long run. To understand the long-term association between these variables, it is important to note that both the exchange rate and consumer prices are determined endogenously and respond differently to various shocks. Estimates of the effect of the exchange rate on consumer prices essentially capture the average response of the two variables to a variety of shocks. To explain the long-run relation between the exchange rate and consumer prices, it is useful to discuss how these variables would respond to different shocks. We can distinguish three types of shocks.

First, there are temporary shocks to the foreign exchange and financial markets. These arise largely from policy interventions and private speculation triggered by changing expectations of future values. These shocks account for much of the short-term variability of the exchange rate, but may have little effect on consumer prices for reasons discussed above.

Second, there are permanent shocks to the money stock. These shocks would be fully passed on to both the exchange rate and consumer prices in the long run. Suppose, for example, that home money supply increases permanently by 10 percent. Assuming that the long-run money demand is unchanged, monetary equilibrium would require that the CPI rise by 10 percent in the long run. As this change would not affect the equilibrium relative price of non-traded goods ([gamma]), relation (4) implies that the exchange rate would also rise by 10 percent. Thus a permanent monetary shock would bring about a change in the price level that would match the exchange rate change in the long run.

Finally, there are real shocks (to technology and preferences) that lead to permanent changes in relative prices. These shocks would influence only the exchange rate in the long run. As an example, assume that labour productivity in traded goods increases permanently. This change would increase the wage rate in both the traded and non-traded goods sectors and increase the equilibrium price for the non-traded goods sector (where labour productivity has not risen). Since the money market is not affected, the CPI would be unchanged. However, relation (4) would require that the exchange rate fall to compensate for the increase in the relative price of non-traded goods. The long-run response to a permanent real shock, therefore, involves a change in the exchange rate but not the price level.

As the above discussion indicates, the long-run association between the exchange rate and CPI would depend on the relative importance of monetary and real shocks. Monetary shocks would tend to be less important in economies where long-run inflation rates are low. The long-run relation between the exchange rate and CPI is likely to be weak in such economies. (11) Identification of this relation would be made difficult, moreover, by the presence of noise introduced by temporary shocks to the exchange rate.

The long-run link between the exchange rate and prices can be related to Purchasing Power Parity (PPP) theory. According to this theory, the exchange rate and the ratio of home and foreign price levels would exhibit the same proportional change in the long run. This implication can be restated in terms of the behaviour of the real exchange rate, defined as the nominal exchange rate divided by the ratio of the price levels (i.e., the real exchange rate equals [S.sub.t][P.sup.*.sub.t/[P.sub.t]). PPP implies that the real exchange rate will revert to a constant level in the long run. In terms of the model with traded and non-traded goods discussed above, it can be seen from (4) that PPP will hold only if the ratio, [gamma]/[[gamma].sup.*], is constant in the long run. As discussed above, however, this condition would not be satisfied if the home and foreign economies are subject to different permanent real shocks.

3. EVIDENCE FOR PAKISTAN

For our empirical analysis we focus on the period since 1982, during which the dollar-rupee exchange rate was no longer fixed. The behaviour of the exchange rate (ER), the domestic consumer price index (CPI), and an index of foreign consumer prices (FCPI) are shown in Figure 1 from the first quarter of 1982 to the second quarter of 2001. (12) FCPI represents a weighted average of the consumer prices for Pakistan's trading partners expressed in US dollars with weights based on Pakistan's foreign trade. Note that this index is influenced by each country's US dollar exchange rate, and thus tends to be more variable than the consumer price series for individual countries expressed in national currencies.

To examine the exchange rate pass-through to consumer prices, we estimate the effect of ER on CPI, using FCPI as the control variable. As Figure 1 indicates, these series exhibit a marked upward trend and appear to be non-stationary. The Augmented Dickey-Fuller (ADF) test indicates that all three series contain a unit root. (13) Estimation of the pass-through relation in levels could thus lead to finding a spurious relation between these variables. The relation in first differences, on the other hand, would ignore relevant information if these variables were co-integrated.

These variables would be co-integrated if the PPP holds and thus the real exchange rate for Pakistan (RER) is stationary. (14) The path of RER is also shown in Figure 1.

There has been a significant increase in RER (or a decrease in the real value of the rupee) during the 1980s and 1990s, and RER does not appear to converge to a constant value (or a deterministic path). (15) The ADF test does not reject the hypothesis that RER (in logs) has a unit root. (16) We assume that the exchange rate and prices are not co-integrated and thus estimate the relation between these variables in first differences.

Estimates of the exchange rate pass-through are based on a regression equation of the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where lags are introduced to allow for gradual adjustment in prices and et is the error term. (17) This equation can be used to derive estimates of the degree of the exchange rate pass-through over different periods. The pass-through coefficient after in periods is defined as the effect of a one unit increase in log ER in period t on log CPI in period t + m, and can be readily calculated from estimates of coefficients, namely the [b.sub.i']s and [c.sub.i]'s, in (5).

[FIGURE 1 OMITTED]

Equation (5) is estimated using quarterly data from 1982:1 to 2001:2. Two estimates of this equation are shown in Table 1: one based on 4 lags for each variable, and the other using the Schwartz criterion to determine the optimal lag length for each variable. In both cases, the effect of ER on CPI is insignificant. Estimates of the short-run pass-through, in fact, are negative. For example, the regression equation using the Schwartz criterion implies that the pass-through coefficient is -0.02 in the current quarter, and -0.01 after 4 quarters. Equation (5) focuses on the effect of the US dollar exchange rate. For a given US dollar-rupee rate, the rupee value in another currency could vary because of changes in the currency's exchange rate with respect to the US dollar. The effect of such a change on consumer prices would operate via FCPI. As Table 1 shows, however, even this variable does not have a significant impact on CPI. (18)

The unresponsiveness of consumer prices to the exchange rate can be seen clearly by looking at the behaviour of the inflation rate after large rupee devaluations. During the sample period, the rupee registered quarterly depreciations of more than 10 percent only three times, once at the beginning and twice at the end of the period. Figure 2 shows the behaviour of the inflation rate for several quarters after these devaluations. As the figure shows, the devaluations do not have any appreciable effect on the path of the inflation rate.

Consumer prices include retail margins that depend on local services and are likely to be insensitive to the exchange rate. It may be thought that the exchange rate may exert a stronger influence on the wholesale price index (WPI) that excludes retail margins and gives more weight to traded goods. Equation (5) was thus reestimated replacing [DELTA]logCPI by [DELTA]logWPI. The effect of ER remained insignificant even in this equation. (19) These results are consistent with the recent view discussed in the previous section that even the traded goods prices respond weakly to the exchange rate.

One limitation of the pass-through relation estimated above is that exchange rate changes are not exogenous, but reflect the effect of a variety of shocks. Each shock may have a different pattern of effect on prices. A structural model is needed to identify individual shocks and to trace the effect of each shock on the exchange rate and prices. Although a thorough analysis along these lines is beyond the scope of this paper, we briefly explore this issue by estimating a simple Vector Auto-regression (VAR).

We consider a VAR with two endogenous variables, [DELTA]logER and [DELTA]logCP1, and one exogenous variable, [DELTA]logFCPI. In this simple framework, we can distinguish between two types of shocks: (1) shocks to asset markets and (2) shocks to goods markets. Asset market shocks are thought to account for much of the observed exchange rate volatility. It is thus interesting to examine how these shocks pass-through to consumer prices. To identify these shocks we exploit the assumption that information about goods markets becomes available with a lag, so that asset market shocks are independent of the contemporaneous value of [DELTA]logCPI. Under this assumption, the reduced-form shock to [DELTA]logER can be viewed as an asset market shock, and Cholesky decomposition can be used to estimate the response of consumer prices to this shock.

[FIGURE 2 OMITTED]

Figure 3 shows the response of [DELTA]logCPI to a one standard-deviation shock to [DELTA]logER over 10 quarters. (20) Two standard deviation bands for this impulse response function are also shown in the figure. The response of the rate of inflation to the shock is close to zero not only in the first year but even after two years. The variance-decomposition analysis shows that shocks to [DELTA]logER explain 97 percent of the variance of this variable after 10 quarters and 96 percent of the variance after 20 quarters. These results suggest that asset market shocks dominate exchange rate changes, and a weak response of CPI to these shocks accounts for the absence of a significant exchange rate pass-through.

Our empirical analysis does not support the results of Ahmad and Ali (1999) that a devaluation has a significant impact on inflation. We believe that their results differ from ours because they estimate a model that is based on some fairly restrictive assumptions. For example, they assume that there is a complete exchange rate pass-through to import prices. This assumption is important for their results, but is not supported by recent theoretical models or empirical evidence. They also estimate relations linking the price level and the exchange rate in (log) levels. As noted above, estimation of relations between non-stationary variables in the level form can produce spurious results. (21)

4. CONCLUSIONS

This paper challenges the popular view that devaluations tend to cause inflation in Pakistan. The empirical analysis in the paper finds no evidence of a significant pass-through of rupee depreciations to consumer prices in the short run. This finding is consistent with recent theoretical analysis that suggests a weak short-run association between exchange rate changes and inflation. It would appear, therefore, that concerns about the inflationary consequences of devaluation in Pakistan are somewhat misplaced. Stability of the nominal exchange rate may be desirable for many reasons, but not because of fears that exchange rate fluctuations will impose an inflationary cost on the economy.

In the last two decades, the rupee has depreciated significantly not only in nominal but also in real terms. This long-period loss of rupee's real value implies that even in the long run, Pakistan's inflation rate has not fully reflected the rate of rupee depreciation. One interesting issue that remains to be explored is: What factors have caused the long-term change in the real exchange rate, and what effects has this had on the Pakistan economy.

[FIGURE 3 OMITTED]

Authors' Note: The paper was prepared while Ehsan Choudhri was Visiting Professor at the IMF Institute. The views expressed are those of the authors and do not necessarily represent those of the International Monetary Fund.

REFERENCES

Ahmad, Eatzaz, and Saima Ahmed Ali (1999) Exchange Rate and Inflation Dynamics. The Pakistan Development Review 38:3,235-251.

Burstein, Ariel, Joao Neves, and Sergio Rebelo (2001) Distribution Costs and Real Exchange Rate Dynamics During Exchange-rate-based Stabilisations. Journal of Monetary Economics (forthcoming.)

Burstein, Ariel, Martin Eichenbaum, and Sergio Rebelo (2001) Why Are Inflation Rates So Low after Large Contractionary Devaluations? (Mimeographed.)

Choudhri, Ehsan U., and Dalia S. Hakura (2001) Exchange Rate Pass-through to Domestic Prices: Does the Inflationary Environment Matter? (IMF Working Paper 01/194.)

Devereux, Michael B., and Charles Engel (2001) Monetary Policy in the Open Economy Revisited: Exchange Rate Flexibility and Price Setting Behaviour. (Mimeographed.)

Krugman, Paul R. (1987) Pricing to Market When the Exchange Rate Changes. In Sven W. Arndt and J. David Richardson (eds.) Real-Financial Linkages among Open Economies. Cambridge, Mass.: MIT Press.

McCallum, Bennett T., and Edward Nelson (1999) Nominal Income Targeting in an Open-Economy Optimising Model. Journal of Monetary Economics 43, 553578.

Siddiqui, Rehana, and Naeem Akhtar (1999) The Impact of Changes in Exchange Rate on Prices: A Case Study of Pakistan. The Pakistan Development Review 38: 4, 1059-1066.

(1) A related view emphasises the role of "input inflation" via devaluation-induced increases in electricity tariffs and prices of petroleum products (ABN AMRO Economic Bulletin, July 2002).

(2) Our results complement those of Siddiqui and Akhtar (1999), which show no causal relation between changes in the exchange rate and consumer-price inflation in Pakistan.

(3) The degree of pass-through to a particular price index is defined as ,me elasticity of the price index with respect to the exchange rate. The pass-through is complete when this elasticity equals one.

(4) Arbitrage would eliminate inter-country price differences under these conditions. This result is referred to as the "Law of One Price".

(5) A 1 percent change in the exchange rate will cause a I percent change in the traded goods price index according to (2) and thus a [theta] percent change in CPI by (l).

(6) See, for example, Krugman 0987) for a discussion of pricing to market. It is assumed that trade costs and other factors segment international markets and make it difficult to arbitrage inter-country price differences.

(7) The price of a traded good i supplied by a foreign producer would equal [[P.sup.T.sub.it] = [[mu].sub.it][S.sub.t][([C.sup.*.subi.t], where [[mu].sub.it] represent the markup and [C.sup.*.sub.it] is the foreign marginal cost. The pass-through would be incomplete if [[mu].sub.it] is inversely related to [S.sub.t]

(8) See Devereux and Engel (2001) for a discussion of local currency pricing and its implications for monetary policy.

(9) A number of recent open economy macro-economic models [e.g., McCallum and Nelson (1999)], in fact, treat all imports as intermediate inputs.

(10) See Burstein, Neves, and Rebelo (2001) for a discussion of the importance of the distribution costs.

(11) Choudhri and Hakura (2001) present evidence that the pass-through (in the short as well as the long run) is positively related to the average inflation rate across countries.

(12) The source of all data is IMF, International Financial Statistics. The series on the real effective exchange rate for Pakistan was used to construct the FCPI measure.

(13) Applying the ADF test to each series expressed in logs, including an intercept, a deterministic trend, and using up to 4 lags, the test statistic does not reject the unit-root null at the 10 percent level for all three series.

(14) 1n this case, there is a co-integrating relation between logs of ER, FCPI and CPI with a cointegrating vector (1, 1, -1).

(15) One possible explanation of the sharp increase in RER is that the traded-goods productivity gap between foreign countries and Pakistan has widened over the past two decades.

(16) The ADF statistic with an intercept, trend, and 4 lags is -2.038 while the 10 percent critical value is -3.160.

(17) For a discussion of a theoretical model that would suggest a pass-through relation of this form, see Choudhri and Hakura (2001).

(18) We also estimated (5) after redefining ER as the effective exchange rate (i.e., the price of a basket of currencies using Pakistan's trade weights), and FCPI as the corresponding foreign consumer price index. The influence of the effective exchange rate was statistically insignificant in this equation as well.

(19) The results are available on request from the authors.

(20) The VAR includes 4 lags of each endogenous variable. A constant term, and the current and 4 lagged values of [DELTA]logFCPI are also included in each VAR equation. The impulse response function is based on a Cholesky decomposition with [DELTA]logER as the first variable.

(21) They claim that their relations are co-integrated. Their co-integration test (an ADF test on the residuals), however, is applied after imposing a number of ad. hoc restrictions on each relation. We eschew estimating our pass-through relation in the level form in view of the indication of a unit root in the real exchange rate.

Ehsan U. Choudhri is Professor of Economics, Carleton University, Canada. Mohsin S. Khan is Director, International Monetary Fund, Washington, D. C.

Table 1
Estimates of the Pass-through Relation

 Coefficient (t-value in Brackets)

Variable (1) (2)

Constant 0.007 (1.29) 0.007 (2.01)

[DELTA]logCP[I.sub.t-1] 0.450 (3.56) 0.444 (3.71)
[DELTA]logCP[I.sub.t-2] -0.300 (-2.23) -0.311 (-2.42)
[DELTA]logCP[I.sub.t-3] 0.275 (2.03) 0.283 (2.21)
[DELTA]logCP[I.sub.t-4] 0.239 (1.89) 0.244 (2.05)
[DELTA]logE[R.sub.t-1] 0.004 (0.06)
[DELTA]logE[R.sub.t-2] -0.027 (-0.45)
[DELTA]logE[R.sub.t-3] -0.033 (-0.59)
[DELTA]logE[R.sub.t-4] 0.065 (1.18)
[DELTA]logFCP[I.sub.t] -0.022 (-0.55) 0.005 (0.16)
[DELTA]logFCP[I.sub.t-1] 0.009 (0.24)
[DELTA]logFCP[I.sub.t-2] -0.012 (-0.31)
[DELTA]logFCP[I.sub.t-3] 0.007 (0.20)
[DELTA]logFCP[I.sub.t-4] 0.043 (1.10)
[[bar.R].sup.2] 0.260 0.307
S.E. of Regression 0.010 0.010

Note: The dependent variable is [DELTA]log[CPI.sub.t].
The lags in regression (2) are determined by the
Schwartz criterion.