Standard Errors in parenthesis

v)

A RATIONAL EXPECTATIONS THEORY OF REALIGNMENTS AND SPECULATIVE ATTACKS: THEORY AND EVIDENCE

Gordon M. Phillips C98 Honors Thesis May 28, 1985

I would like to thank and give credit to Professors Ian Domowitz and Robert Flood for consultation. Specific citation is given for many of the parts of the model tested. I would also like to thank Professor Sullivan for the GQ-OPT tape and Tom Doan for fortran and IBM VM help. I would also like to thank Jack Pressman for help in obtaining a substantial part of my data.

A RATIONAL EXPECTATIONS THEORY OF REALIGNMENTS AND SPECULATIVE ATTACKS: THEORY AND EVIDENCE

ABSTRACT:

The first question one might ask concerning this paper is why focus on fixed exchange rate regimes. But as late as 1983, 8 9 countries were classified as having a fixed regime. 38 countries are pegged to the dollar, 13 to the French franc, 14 to SDRs, and 24 to

different baskets of currencies. As implied by these facts, fixed exchange rate regimes are still very common. This study will focus on the European Monetary System (EMS), a group of 8 European

countries that are fixed against the European Currency Unit (ECU), a composite monetary unit containing fixed quantities of each EC currency. My study will examine the stability of fixed exchange rate regimes when exchange rate policy is endogenous - evident when countries pegged to each other have divergent monetary policies.

The goal of this study is to build a model that predicts when realignments of the central parities will occur.

The first section of this paper presents the underlying

theory. Section two presents the background of the EMS showing why one would believe the theory would model the situation. Section three develops a model which predicts the probability of

devaluation next period and does not assume purchasing power

parity. Section 4 presents the estimation methodology with Section 5 presenting empirical results from five countries inside the EMS.

The last section gives a discussion of these results.

SECTION 1: SPECULATIVE ATTACK THEORY

The theory that lies behind this paper can be termed speculative attack literature. This area of literature was developed by Salant and Henderson's (1978) analysis of exhaustible resource pricing. Krugman (1979) first applied this idea to exchange rate

crises. Recent extensions include Salant (1983), Flood and Garber (1984), Obstfeld (1984), Grilli (1985). There are two empirical pieces by Blanco and Garber (1984) and Collins (1985).

The central point that is made by this literature is that exchange rate policy in a fixed regime is frequently endogenous rather than exogenous. A government's fiscal policy and its implied deficits, funded by domestic credit creation are more accurately primary exogenous variables. To maintain equilibrium and the fixed rate, countries would have to subordinate their monetary policy to the country or countries they are pegged against, which many countries do not want to do. A country is supposed to exchange currency, whenever requested at a set rate.

Without equality in the balance of payments, a country may be

forced to deplete reserves of foreign currency that it holds. When these reserves are extinguished, a country is forced to devalue its currency or go on a floating system, where the value is determined by supply and demand. In a system such as the EMS, where countries have to remain inside a symetrical band, a country could also be

forced to revalue as the value of the currency could remain at the bands lower band.

As one can see, agents can form rational expectations about

this devaluation or revaluation, and if they convert currency before the realignment, large profits can be made. These

expectations and the action taken obviously can cause the fixed rate to collapse "early".

Agents will buy foreign currency when

e(t+l)>e where: e: floating rate post attack e: fixed exchange rate

ie. when the rate after the collapse is higher (currency is worth less) than under the fixed rate. Agents will sell currency when:

e(t+l)<e same notation as above.

This floating rate is determined by market fundamentals. It is worth more than the fixed rate in a stable fixed regime. To help temporary fluctuations, countries hold reserves of foreign

currencies to balance out temporary movements in the BOP.

A regime can run into problems when they cannot support growth in monetary stock, with their reserve currency. If such growth is expected to persist, people's expectations can hasten the

currency's collapse. The converse problem also occurs when too much currency is accumulated. When there is some type of symetric band, a country could be forced to revalue and again people's expectations could hasten this collapse.

One of the early papers in this area is by Paul Krugman

(1979). The psychology of investors is important, Krugman says, in deficit financing. The extent to which a government finances the deficit by running down its foreign reserves is determined by the private sector's willingness to acquire additional domestic money.

Rather than picking an exchange rate rule and holding to it, domestic fiscal policies and implied government deficits are the primary goals of the government. Thus one of the central parts to

any model is making exchange rate policy endogenous and the central banks domestic credit creation policy exogenous. Krugman's model is a simple one with purchasing power parity, fully flexible wages and prices. Investors participate in a "speculative attack" on the government's reserves by changing the composition of their

portfolios, protecting themselves against currency devaluation by purchasing foreign currency from the government. If investors' expectations are rational, the government's reserves are liquidated by a speculative attack. Thus their actions are self fulfilling and the simple expectation actually produces the attack.

Flood and Garber (1984) add to the model the important concept of an underlying floating exchange rate. This "shadow rate" is unobserved by actors but can be predicted because it is based on market fundamentals. They use two examples to portray exchange rate crises. First they use a continuous time model where the actors have perfect foresight. In the second example they use a discrete time model with uncertainty. With perfect certainty the currency never produces a foreign discount. As soon as agents foresee the collapse to a higher floating rate they purchase the government's stock of foreign reserves at the point where losses are avoided and collapse the system. Under perfect certainty this point occurs when the shadow floating rate (e) is equal to the

current fixed rate (e) - ie. e = e. It is never profitable for the agents to purchase reserves when e < e. They would obviously lose money. The attack will thus occur with perfect certainty when e = e (at time t=T). Profits would not be made, but losses avoided.

Graphically:

e

underlying flexible exchange rate

time

Underlying the depreciation of the true value of the currency would be an expansion of domestic credit through domestic credit creation or balance of payments deficits. Flood and Garber solve for T, the time of the collapse of the fixed exchange rate regime.

They then bring in uncertainty. This uncertainty concerns

information about the growth of domestic credit, domestic policy, and interest rates, accounting for the forward discount of a

currency. It is based upon the probability that the currency's value will go down. The probability of a collapse at t+1 is the probability that e(t+l) > e(t). This probability is given as the probability density function of an exponential distribution. The forward discount rises exponentially with the level of domestic credit until the currency collapses.

An extension of this study is made by Obstfeld (1984) and empirically by Blanco and Garber (1984), including the possibility of a devaluation. In this case the expected rate of change must

always meet or exceed the shadow floating rate. The analysis by Obstfeld models the range of decisions open to a government facing a shortfall of foreign reserves. His model allows possible

devaluations - repegging after a transitional period of floating.

He notes that typically foreign reserves will not be exhausted at the time of collapse - extending the idea of Flood and Garber.

The demand for real money balances is modeled as a decreasing

function of the expected rate of exchange rate depreciation. The nominal money supply is at a constant factor of the exchange rate, but the central bank causes domestic credit to increase at a

constant rate. This increase decreases foreign reserves and causes the real value of the currency to depreciate. The

determinants of the time of collapse also include the length of the period of transitional floating.

Blanco and Garber (1984) develop a model which is able to be tested empirically. Their model provides the basis for my study.

It gives the probability a fixed rate regime will devalue one period ahead and the expected value of the new fixed rate

conditional upon a devaluation. The probability of devaluation and exchange rate policy are endogenous variables with the rate of

domestic credit growth or domestic policy being the exogenous

driving force. In the model they use domestic credit growth and a money market equilibrium to determine the path of the process (h^) which drives the underlying flexible rate (e). Thus the

probability of a devaluation exists at all periods as the shadow flexible exchange rate is present at all times during a fixed regime.

The rule for establishing a new fixed rate is that a viable new fixed rate ("e) must be greater then the underlying floating

rate, e > e . Thus e places a lower bound on the value of a new fixed rate. Continuing with the idea brought out by Krugman,

reserves do not necessarily have to be exhausted. Foreign currency reserves might be but the government can also borrow against gold reserves. Thus the key step is to derive the underlying floating rate based on market fundamentals and then to determine minimum reserves under which the authorities will still defend the current fixed rate regime. Blanco and Garber get results for the Mexican experience from 1973 - 1982 quarterly. Given the simplifying assumptions they made the model seems to do well.

Testing their model out on the EMS data, I found that it did not work at all in its present format. The standard deviation of the residuals of the h(t) process was a factor of 10 smaller for the countries in the EMS than Mexico which caused the negative

exponent of the E[et+-j] to expload so that the expression would not evaluate. (In fortran terminology - exponent underflow exception).

Grilli (1985) using this model for investment banking contracts and gold reserves also finds that the standard deviation of the

residuals of this h(t) process is also a factor of 10 smaller.

I then changed several of their simplifying assumptions. First they have a poorly specified money demand model. They note this fact themselves observing the Box-Pierce statistic. This money demand model is instrumental in determining parameters for the underlying flexible exchange rate expression. Specifically the standard deviation of the residuals on the stochastic disturbance which determines the new fixed rate and the expected rate in period t+1 might not be accurate given this money demand specification.

The second simplifying assumption is purchasing power parity (PPP).

This assumption does not seem valid if goods prices adjust slowly.

The factors causing trade inbalance could persist causing pressure on the exchange rate. Also price differentials might be expected to persist when the shares of domestic goods in domestic and

foreign consumption are not equal. Even after a realignment

continual pressure may exist because of stickiness of prices when domestic credit creation in excess of the trading partners.

Grilli (1985) adds to the speculative attack literature in a theoretical model which also includes the possibility of

revaluation. This article is especially relevant for testing in the EMS because of strong currencies like Germany's.

My model specifically addresses the above criticisms of the Blanco and Garber paper. I respecify the money demand function and provide a specific representation of possible divergence from

purchasing power parity. Prices are represented as "sticky", being set in preceding periods at a level expected to clear the demand market. Thus price setting is anticipatory. This model was developed in a theoretical paper by Flood (1981). Evidence from the EMS shows that PPP does not hold well implying that the

anticipatory pricing model might be more appropriate. This anticipatory pricing model allows for PPP divergence since

consumption weights need not be equal across countries [see Flood (1981). I also include the probability of revaluation as

developed theoretically by Grilli. After putting together the model I proceed to test it empirically for five countries in the EMS: Germany, France, Denmark, Italy, Netherlands. Germany and the Netherlands are examples of countries with strong currencies, which have experienced revaluations. Italy, France, and Denmark are countries which have experienced varying degrees of

devaluations.

SECTION II: BACKGROUND OF THE EMS

The European Monetary System (EMS) is an ideal system to test a complete model of speculative attacks. Because of the divergence in currency strength we see examples of currency devaluation and revaluation. There has also been extreme divergence of domestic credit growth and the lack of PPP. The following summary tables show the existence of these two pressures. The tables show

evidence one group of courtries with a more expansionary monetary policy. The realignments of the currencies also support the two groupings; the countries with the more expansionary monetary policy have experienced the devaluations.

Growth Rates of Money Stock (Ml) in EMS countries

C o u n t r y L212 L2SD 1M1 1M2 a v e r a g e 1 9 7 9 - 8 2 2 . 0

2 . 1 2 . 7 1 1 . 0 1 2 . 3 1 6 . 0

Inflation rates (CPI) in EMS countries

Country 1979 1980 1981 1982 average 1979-82 Belgium 4.5 6.6 7.6 9.0 6.9 Germany 4.1 5.5 5.9 5.4 5.2 Netherlands 4.2 6.5 6.8 6.3 5.9 Denmark 9.6 12.3 11.7 10.0 10.9 France 10.8 13.3 13.3 11.8 12.3 Italy 14.7 21.2 17.8 16.6 17.6 Belgium

Germany Netherlands Denmark France Italy

2.5 2.9 2.8 9.9 11.8 23.7

.2 3.9 6.0 10.9 6.4 12.9

2.2 -1.6 -2.4 11.8 15.9 9.8

3.3 3.2 5.0 11.4 14.8 18.0

Source: Commission of the European Communities, The EMS, (1984) Thus this summary data argues that the central fixed parities would not be stable. The following also shows evidence of divergence.

History of Devaluations and Revaluations from 1979-1985.

The following currencies are abbreviated in this table: Belgian franc (BF), deutshe mark (DM), Danish krone (DKr), French franc

(F), Irish pound (Ir), Luxenbourg franc (LuxF), Netherlands guilder (G), and Italian Lire (Lit).

1. September 24, 1979 - Upward shift in cross rate between DM and DKr of 5%. Shift in cross-rate between DM and other currencies of

99-

2. November 30, 1979 - Devaluation of Dkr by 5% against other currencies.

3. March 23, 1981 - Devaluation of Lit by 6% .

4. October 5, 1981 - Revaluation of DM and G. by 5.5% against DKr,BF, LuxF, Ir, . Devaluation of F, Lit by 3% against DKr, BF,

Ir.

5. February 22, 1982 - Devaluation of BF, LuxF by 8,5% and of DKr by 3%.

6. June 14, 1982 - Change in bilateral rates: between F and DM,f.

10%, between Lit and DM, G - 7%, between DKr, BF, LuxF, Ir, and DM, G - 4.25%.

7. March 21, 1983 - Change in central rates: revaluation of DM 5.5%, G - 3.5%, Dkr 2.5%,BF and LuxF both 1.5%; and devaluation of the F and Lit 2.5% and Ir 2.5 %.

8. July 22, 1985 - Devaluation of the Lire 8% against all other currencies.

Source: commission of the European Communities and Fund Staff.

An additional factor that makes the EMS a good example for a

rational expectations model is the construction of the exchange rate rules for the EMS. Currencies float around a central band consisting of its value in terms of European Currency Unit (ECU).

Countries are required to keep their currency within 2.5% of the central parity, 6% for Italy. This requirement means that if a country is not willing to subordinate its domestic policy to that of the other countries, the country will be forced to change the value of its currency. The weak currency will obviously feel much more pressure as it would not be able to trade eventually as its

foreign reserves become depleted. The previous divergences in domestic credit growth and interest rates show evidence of this unwillingness. In the EMS, countries are forced to remain within this band because bilaterial central rates are maintained between participating countries. These are enforced through compulsory and unlimited intervention on the foreign exchange market. Between two countries, the central banks are required to intervene. Also the observance of limits is to some extent guaranteed by the market, because the monetary authorities are required to meet the market demand for the currencies within the established bands.

Section 3: The Model

Central idea is that underlying a fixed rate regime there is a unobserved floating rate that would prevail if the fixed rate collapsed. Actors can cause a fixed regime to collapse by

depleting foreign reserves. Rational actors would always do this when the new floating rate is higher than the old fixed or also obviously when a devaluation is expected.

ie. when e(t) > e(t), a buying attack would occur,

A new viable fixed rate,e(t) > e(t) the underlying floating rate.

Thus e(t) places a lower bound on the value of a new fixed rate.

Thus the key step is to derive the underlying floating rate based on market fundamentals and then to determine the minimum reserves under which the authorities will defend the regime.

Under an exchange rate regime that pegs within a band a symetrical probability of revaluation could also occur for countries whose currency was undervalued. This situation occurs for the EMS as brought out earlier. Rational agents that expect a revaluation would participate in a "selling" attack when the value of the currency after the revaluation would be greater than under the fixed peg.

A, —

ie. when e(t) < e(t), a selling attack would occur.

The monetary authorities would then accumulate foreign reserves until a critical maximum reserve was reached. The existence of

this maximum level of reserves is dependent on the existence of the symetrical band that the countries have to stay within, otherwise the country could theoretically continue to accumulate reserves.

There are reasons even in this case not to continue to accumulate though.

4 central equations of the money market.

(1) m(t)-7t(t) = B + Qy(t) + ai(t) + a2CURR + a3lNF + ^i^t. (Real money Demand)

(2) m(t) = R(t) + D(t). (Money stock = domestic money + foreign reserves)

(3') 7t(t)-7c*(t)=(0-0*)( co + ci m(t-1) + C2i*(t-1) -e(t)) + e(t)

Old equation # (3) 7c(t) = 7c*(t) + e(t). (Purchase Power Parity)

(4) i(t) = i*(t) + E[e(t+1)] - e(t) (uncovered interest parity)

where e= log(exchange rate) y=log(GNP)

7t = log price index,

i = nominal interest rate

CURR= expected rate of currency depreciation INF= expected inflation rate

m = log money stock

0 = share of domestic goods in domestic consumption 0 =share of domestic good in foreign consumption.

= indicates foreign variable

R = minimum level of foreign reserves

Equation 1 gives the demand for real money balances as a function of log income, nominal interest rates, expected currency

depreciation (CURR), and expected inflation (INF). The last two terms are alternative costs of holding money obtained through

instrumental variable estimation. The instruments used were lagged inlation and lagged interest rate. This specification is from

Domowitz (1985). Equation 2 decomposes the domestic money stock

into reserves and domestic credit. Equation 3 allows for a divergence from purchase power parity. The first part of the first term, (0-0*) accounts for differences in import consumption and the rest for possible sticky price movements for domestic

traded goods. This equation follows Flood (1981). Equation 4 is uncovered interest parity, not allowing for a possible risk

premium. Exchange rates are in logs and interest rates are nominal.

Derivation of new equation (3) - follows Flood (1981)

From:

7C(t) = 0p(t) = (l-0)q(t) and

7C*(t) = 0*(t)p*(t) + (l-0*)q*(t)

we get:

7C(t)-JC*(t) = (0-0*) (p(t)-q(t)) + e(t) 7t = price index.

0 = share of domestic goods in domestic consumption.

0 =share of domestic good in foreign consumption p(t) = relative price of domestic traded goods.

q(t) = relative price of foreign traded goods,

normalizing q* to unity in levels and taking log gives q*=0 from q(t)= q* (t) + e(t)

we get:

(3a) 7C(t)-7C*(t) = (0-0*) (p(t)-e(t)) +e(t)

Substituting in for price level from an anticapatory pricing rule.

Prices are set so as anticipated demand = supply - level expected

to clear the goods market.

E, ,[y(t)] = E^{t - 1} [y (t) ] This allows for sticky prices.

where -

yd( t ) - fi0 - B1< pt - qt ) - fi2(it-Et[7Ct+1-7Ct] + wfc* e x o g e n o u s f o r c i n g p r o c e s s e s :

mt = Pmt - 1 + ^t

1

t = H-i

^t

* ~k -k

wt = wt_1 + \|/t

*

qt = 0 (normalized to zero.)

Solving using method of undetermined coefficients with exogenous forcing processes, standard money demand equations (1) - (3) as in the main model, and using anticipated pricing rule - E, , p. =

pt and Et_ , pt = P+- • w© get:

(3b) p(t) = c0 + c1 m(t-l) + c2i*(t-l) with C Q = constant

cx - p / ( 1 + a(l-p) ) c² = a

Substituting (3b) into (3a) gives equation (3) which is used in the model:

(3) 7c(t)-rc*(t)=(0-0*)( c0 + ci m(t-1) + c2i*(t-1) -e(t)) + e(t)

Now solving from equations 1-4:

(5) -aE[e(t+1)] + [1+a +(0-0*) ]e(t) = log[D(t) + *Rexp(e)] - p*(t) -3 - Qy(t) - ai*(t) - (0-0*)( c0 + c-| m(t-1) + c2i*(t-1)) - nt

Construct h(t) = -aE[e(t+l)] + [l+a+0-0*]^ (t)

where h(t) = composite underlying forcing processes

Assuming that the actual process is AR(1) and that the parameter estimates in h(t) are consistent estimates of the actual process.

(Note that setting the constructed estimates of the parameters equal to the AR(1) process ignores the fact that these are estimates with a distribution around the true value.)

then

(6) h(t) = d1+ d2h(t-l) + v(t)

AR(1) forcing process exogenous to the exchange rate with v(t) white noise N(0,1 ) o

As noted by Blanco and Garber, the exogeneity of this ht process is a strong assumption since variables such as income and domestic credit may effect it. But then the log - linear money demand function could not be used.

Solving the difference equation in h(t) for e(t). (from Grilli-1985)

et - ( 1/ [l+a+0-0*]) A>(a / [l+a+0-0*]) Etht +.

Assuming that ht follows the AR(1) process given above, and apply

t h e W i e n e r - K o l m o g o r o v p r e d i c t i o n f o r m u l a :

Etht + j= ( d1/ l - d2) - [ d1( d2)j / l - d2] + ( d2)jht j

s o :

et = ( 1 / [l+a+0-0*]) S0(a / [1+a+o-0*])[ (di/1-d2) - [d1(d2)J /1-d2] + (d2)ihtJ]

•J

Thus the solution is:

(7) e ( t ) = iiad-L + | l h ( t ) w h e r e |i = l / [ ( l + a ) + 0 - 0 * - a d²]

assume after a devaluation the new fixed rate e1( t ) = e(t) + bv(t) The rule states that after realignment the central bank will set a new fixed rate equal to the underlying floating rate (+) or (-) a quantity dependent on whether the currency is being devalued

because of lack of reserves or revalued because of excess reserves. ie. b>0 for deval., but for a revaluation b<0.

From this point on the devaluation side of the model is developed, with the revaluation side having a parallel development but the probability of a revaluation being :

(8) Pr(t+l|t) • Pr( et + 1 < i~t+1 ) . (Grilli 1985 develops this theoretical side)

The probability of a devaluation in period t+1 in period t is:

A ~

(8) P r ( t + l | t ) = P r ( et + 1 > et + 1) whxch e q u a l s :

(9) = P r ( u h ( t + 1 ) + j i a d1 + b v ( t + l ) t " et + 1) i e . et + 1 > e "t + 1-

(10) Pr(t+l|t) = Pr(v(t+1) > K(t)) - 1- F(K(t)) F(K(t)) is the cdf assoc. with g \v) where g(v) is the normal density

function.

One can compute the one step ahead devaluation probabilities after the parameters in K(t) are estimated,

where:

K(t) = [l/(n+b)][e- |iad1 - u(dx + d2h(t))]

Agents can form future exchange rate expectations from the average of the current fixed exchange rate and the rate expected to

materialize conditional on a devaluation both weighted by the respective prob. of occurrence. These exchange rates are expectations because they only have estimates of the true parameters.

(ll)E[e(t+l)] = [1 - Pr(t+1 I t)]e + Pr(t+l| t)E[e(t+l) |v(t+l) > K(t)]

where the conditional expectation is:

(12) E[e(t+1) | v ( t + l ) > K ( t ) ] - H d i ( l + a ) + (id2h(t) + [(I + b ] E [ v ( t + l ) | v ( t + l ) >K(t) ] w h e r e E [ v ( t + 1) | v ( t + 1) >K ( t ) ] = f v g ( v ) / [ l - F ( K ( t ) ) ] d v .

thus through substition and using the normal density function, one can get an expression for unconditional forecast of the exchange rate for t+1

(13) E[e(t + 1)] = F ( K ( t ) ) e + [l-F(K(t) ) ] t|^d1 (1+a) + ud2h(t)]

+?i(|l-b)exp[-.5((K(t)) A )2] / >£TT

given that f(t) = E[e(t + 1)], where f = forward rate construced through interest rate arbritrage, minimize

(14) [f(t) - E[e(t+1)]] ie - sum of squared residuals to obtain estimates of R and b.

Section 4 : Estimation Procedure.

The estimation procedure used is an iterative process using

Quandt's GQ-OPT and instrumental variable OLS. Quandt's GQ-OPT package contains several non-linear unconstrained optimization

fortran algorithms designed to optimize general non-linear

functions that are supplied by the user. Depending on the type of function there are several possible methods available in GQ-OPT which will find a local maximum or minimum of this user-supplied

function, in my case equation #14 [f(t) - E[e(t+1)]] . Inflection points are avoided by checking second derivitives. Davidon,

Fletcher, Powell (DFP) and GRADX are used which are good for quadriatics. I had to write two fortran programs 1) a calling

program and 2) a subroutine to define the function to be optimized.

All variable were in double - precision to obtain further accuracy.

The methods are quadriatic hill-climbing routines which evaluate the gradient of the function at each step to determine the next step to take.

The following steps show the iterative procedure which was used.

Money demand parameters were estimated independently of the rest of the parameters.

1. Obtain money demand parameters, (£>,Q, a, a2, ag) for equation #1 - instrumenting out inflation and currency depreiation (INFL,

CURR). The instruments used were inflation rate and interest rates lagged three periods. Next obtain price parameters for equation (3b) p(t) = CQ + c-^ m(t-l) + c2i* (t-1)

through OLS - (CQ/C-WCOX Note consistent parameters are

obtained, even in the presence of serial correlation, given that the independent variables are exogenous, which is why instrumental variable estimates are used for inflation and currency

depreciation.

2. Pick initial estimates of R and b through a two dimension Grid Search which evaluated the function 2500 times constructing a 50 X 50 contour plot. The contour plots are in Appendix 1. Next step is to construct h(t)= ~h(t) with these initial estimates.

3. Estimate cL , cU / ^ from h(t) = d-, + doh(t-l) + v(t) using OLS giving initial estimates of these parameters.

4. Plug these in E[e(t+1)] constructing [i and minimizing equation

#12 : [f(t) - E[e(t+1)]]2 to obtain estimates of R and b.

This step requires nonlinear estimation minimizing the sum of square residuals. I used GRADX and the Davidon-Fletcher-Powell algorithm in Quandt's GQ-OPT package. The printouts of the results are in Appendix 2.

5. Return to 2.

Section 5: ESTIMATION

This section shows how variables were constructed and data was obtained, in leiu of a data appendix. Table 1, on the third page of this section, gives the parameter estimates from the money

demand equation #1. Table 2 gives the parameter estimates from the price equation that is put into equation #3 to account for the

absense of purchasing power parity. Table 2 also gives the

estimates for the parameters in the ht process. These estimates are conditional upon the value of the minimum reserves, R. Table 3 gives the estimates from GQ-OPT that are iteratively obtained

conditional upon the values of parameters in the h. process in

Table 2. Again the estimation technique is described in Section 4.

Monthly data from April, 197 9 - June, 1985 was used for the following estimation. Accounting for the use of lags and

differences, there were 71 observations for each variable. The data was defined and obtained as follows:

1. All foreign variables: Given the unique situation of the EMS countries being pegged against each other, the foreign variables were constructed as weighted averages of all the countries in ECU

"currency". They were weighted by exchange rates and country

weights in the ECU. The country weights take into account the size of GNP and international trade importance. Thus the weighting

scheme is analogous to common ways that are used to weight data.

This scheme in effect constructs a new country - the country of

"ECU".

2. Price Indices - obtained from IFS tapes. The tapes allowed me to avoid the problem of periodic adjustments of series which are

page - 21

never published for historic data. The WPI was used to construct real exports and real international reserves. The CPI was used to construct real GNP and real domestic credit.

3. Interest rates - nominal interbank rates for 3 month periods.

From Eurostatistics - Monthly Data for Short Term Economic

Analysis. For the other measures of the cost of holding money - inflation and currency depreciation, I used monthly percentage changes in the CPI and exchange rates respectivlty.

4. Monthly GDP rates - Used 1980 GDP in the country's currency as the base to construct monthly GDP from an industrial production index.

5. Exports and Imports - used to construct 0 and 0 . These were obtained in monthly data off of the IFS tapes and weighted by the WPI of the Country and the WPI constructed for the EC as a whole respectivly.

6. In the estimation, fixed central rates (currency/ECU) were used for the exchange rate. In effect this fact ignores the existence of the bands. At this time including the bands was beyond the scope of this author. I believe the model still captures specualative money demand, and that all parameters are correct. The probabilities of devaluation are much lower than I believe should be the case, but I believe that the shifts should be proportional, thus the real probabilities of devaluation should

just be a constant factor greater because the bands do not flucuate around the central peg.

Estimation results: TABLE

IV ESTIMATION - Inflation (INFL) and Currency deprecation (CURR) were instrumented out using inflation and interest rates lagged 3 periods.

Equation estimated:

(1) m(t)-?c(t) = B + Oy(t) + ai(t) + a 2 C U R R + a3lNF + |it.

Results

DENMARK

-.909 (.201) 1.36 (.113) .058 (.064)

-2.75 (1.80) -.153 (.485) -.182 (.485) -.107 (.482) -.099 (.485) -.134 (.492) -.090 (.483) -.198 (.486) -.174 (.481) -.146 (.482) -.144

(.488) -.159

<:???

(.475)

COUNTRY

FRANCE

-1.40 (.135) .535 (1.11) .00416 (.032) .233 (.736) 6.64 (.95) 6.61 (.95) 6.62 (.95) 6.61 (.95) 6.60 (.95) 6.61 (.95) 6.62 (.95) 6.61 (.95) 6.58 (.95) 6.58 (.95) 6.59 (.95) 6.72 (.95)

I T A L Y

-.257 (.127) .172 (.096) -.226 (.018) -4.05 (.828) 5.58 (-52) 5.55 (.52) 5.57 (.52) 5.58 (.52) 5.56 (.52) 5.58 (.52) 5.59 (.52) 5.56 (.52) 5.58 (.52) 5.59 (.52) 5.55 (.52) 5.62 (-52)

G E R M A N Y

-.735 (.158) .527 (.101) .0688 (.042) .242 (1.59) 2.93 (.483) 2.94 (.484) 2.92 (.483) 2.93 (.49) 2.94 (.49) 2.96 (-49) 2.95 (.49) 2.95 (.49) 2.94 (.49) 2.94 (.49) 3.02 h .49) 2.99 (.49)

N E T H E R L A N D S

-.799 (.155) .663 (.083) .097 (.046) .296 (1.72) 2.23 (.28) 2.23 (.28) 2.23 (.28) 2.25 (.28) 2.31 (.28) 2.32 (.28) 2.26 (.28) 2.26 (-28) 2.26 (.28) 2.25 (-28) 2.25

^28) 2.24 (-28) I N T E R E S T - ( a )

INCOME -(Q) Aet - (a2) ACPIr $3) JAN

FEB MARCH APRIL MAY JUNE JULY AUG

OCT NOV

NOTE

Standard Errors in parenthesis

page - 23

TABLE 2

ESTIMATES OF ADDITIONAL PARAMETERS FROM OLS GIVEN MONEY DEMAND PARAMETERS

(From final run of GQ-OPT parameters Rbar and b.)

Equations estimated:

(3') 7c(t)-7c*(t)=(0-0*)( c⁰ + c-i m(t-1) + c²i*(t-1) -e(t)) + e(t)

(6) h(t) - d±+ d²h(t-l) + v(t) with: v(t) N(0,?i2)

COUNTRY

ITALY

DENMARK

NETHER.

GERMANY

e„

-12.58 (.373)

-3.17 (.201)

-4.17 (.277)

-6.09 (.497)

$

1.04 (.0285)

.681 (.034)

.957 (.058)

1.08 (.084)

_£,

1.75 (.349)

2.31 (.513)

1.68 (.347)

2.13 (.401)

A

Dl .203 (.082)

.0108 (.042)

.0082 (.043)

.0775 (.078)

A D2

.879 (.047)

.984 (.028)

.990 (.025) .971

(.0267) A

X

.061590

.054118

.03449

.040275

FRANCE -6.02 .943 .236 -.0082 .970 .04422 (.178) (.024) (.260) (.0054) (.021)

NOTE: Standard Errors in 2nd line in parentheses above. 1st line contains parameter estimates.

TABLE 3: RESULTS FROM GQ-OPT

ESTIMATES OF MININIMUM RESERVES REQUIREMENTS IN BILLIONS OF NATL CURRENCY AND

COUNTRY

initial parameters:

NETHERLANDS DENMARK ITALY FRANCE GERMANY run two:

NETVERLANDS DENMARK rTALY FRANCE GERMW

.0344812 .060546 .0658819 .03570513 .0484265

.03448412 .05431923 .06150080 .03465253 .04042051 run three-Final run:

NETHER t-stats DENMARK t-stats

rTALY t-stats

FRANCE t-stats GERMANY t-stats

.03448860

.05411755

.06159036

.04422016

.04027574

SHOCK DIST.

ESTIMATES FROM EQUATION #13)

from grjd search from^dst run of GQ-OPT initial (R. b) 1st(R.&

(100,9.5) (150,7.5) (9000.0, 19.5)

(1000, 7.0) (-1.996404 (100,1.5) (2150.26,-

(78.11562,-6.9158) (39.66575,9.1289) (807.654, 38.075) 15.5865)

10,756) 2ND(R.B)

(81.6331,-6.7418) (37.70979,9.88917) (686.9761,40.17292) (-86.66057, 17.66888) (1517.41737,-12.4264) 3RD (Reserves. Shocks (81.59075,-6.74127) (4.692, -7.097)

(37.765, 9.927448) (.8089, 3.093) (666.5395, 40.078) (8.776, 16.0064) (-84.5845, 13.749712) (-.77644,4.5198)

(1517.74185,-12.44079) 1.1606, -3.5679)

page - 25

Section 6: CONCLUSIONS AND DISCUSSION OF RESULTS

In discussion of results, many authors try to convince readers that the model fits reality. In this case this is not possible.

The model presented in this paper is highly stylized and is tested in one form across all five countries. Looking at the probability graphs following, I have to admit that while some meaning can be found, it is not easy.

I will make some brief summerizing comments starting with the parameter estimates. In table 1 several facts emerge. Interest rate elasticity is negative, as theory and common sense tells us it should be. Except for the Netherlands, inflation is significant as an alternative measure of the cost of holding money while currency depreciation is not. There also does not seem to be much

seasonality of money demand in EC countries. Correcting for inflation might account for part of this fact, to construct real variables. Table 2 contains additional parameters conditional upon the value of reserve limits. X is the standard deviation of the ht process and much of the estimation relies on this parameter.

It is used to standardize Kfc for the normal CDF. My results for X agree favorably with Grilli (1985) but are a full factor of 10

smaller than Blanco and Garber. The increase in this parameter, as I experimented with, does give more favorable results for the

probability numbers. The results in Table 3 are based upon all of the other parameters in the model being estimated first, then

iteratively reestimating.

The final values for b are especially encouraging because b

was estimated independent of sign and the sign came out as the

theory would suggest. Including the revaluation cases had not been estimated before to my knowledge and the results were favorable: b was positive for the devaluation cases and negative for the

revaluation cases. b also was significant for all the countries in the analysis.

There are mixed results for the reserve limits , R. France and Denmark did not have significant parameter estimates and the

magnitude of R for them also seems out of line when compared with the actual reserve movements as shown in the graphs of reserves following . On the other hand, Germany, Italy and Netherlands had significant parameter estimates and they seem feasible when

compared with actual reserves.

The probability of devaluation graphs following, chart the probability against time. The eight realignments are indicated across the x - axis. The dashed lines horizontal lines indicate if the country was involved in that realignment - since only certain countries were involved in each realignment. The exact size of these changes are given in a table in Section 2. Looking at the probabilities, one point I want to mention again is that this analysis was run with fixed rates, not the float within the band, so divergence was noted. One might say that the probabilities presented in the graphs are lower that seems right, and I would agree. But I believe the actual movements with respect to each other are correct. One reason for the low probabilities is that the bands in the EMS are not taken into account. The existance of the bands means the country must get in line before any real

problem occurs, thus increasing the probability of revaluation as the edges of the band are approached. Modeling the influence of

page - 27

the bands explicitly would be the next logical step in this model.

I believe including the influence of the bands would improve the predictiveness of this model and make the model a closer fit to reality. The biggest problem with including the bands was a extreme lack of time, the rest of the analysis was already extremely time consuming. Another possible reason for the low probabilities was the relative speed of decisions to adjust the parities. I quote "A common feature of thes readjustments was the calmness and the swiftness with which they were carried out,

possibly reflecting a greater willingness on the part of the

system's member countries to accept limited adjustments when they seemed necessary, instead of accumulating delays as happened too often under the Bretton Woods regime." (Commission to the Economic Council, March 1984)

In conclusion, the model presented does very well considering that it is very stylized and is trying to fit realignments in a very complex environment in one form across all five countries.

Reserves and probabilities seem to be within reason for three out of five countries tested: Netherlands on the revaluation side, and Italy and France on the devaluation side.

P R O B A B I L I T Y OF REVALUATION G E R M A N Y

O

0.1 3 0 . 1 2 0.1 1 0.1 0 . 0 9 0 . 0 8 0 . 0 7 0 . 0 6 0 . 0 5 0 . 0 4 0 . 0 3 0 . 0 2

0.01 I I I 11 I I I I 11 I I I I 11 I I I I I I f I I I I 11 I I I

1 £ 3 4* 5 6* 7 * 6

l l l l I I I l I l l I l l I l l I l I I I l l I I l I l l l l l I I I

indicates involvement in realignment

ACTUAL VS. MAX. RESERVES - GERMANY

i

CD 41 Q . b 41

si

= o

e£

I

41

41

i i I i i i

t i m e / r e a l i g n m e n t #

P R O B A B I L I T Y OF REVALUATION N E T H E R .

Si O

3 1* 5 6*

t f m * X r * a l l g n t n * n t *

"7? indicates involvement in realignment

I I I I I I I I I I I I I I I I I I I I I I I I

7* 8

ACTUAL VS. MAX. RESERVES - NETHER.

LD to .*!

15.

si

= 0

£b

I w

I

41

t i m e / r e a l i g n m e n t #

P R O B A B I L I T Y O F D E V A L U A T I O N

0 . 0 6

T A L Y

0 . 0 5 5 -

^ • f m * / ^ * c ^ l ^ g n f r ^ * n t • *

^ i n d i c a t e s i n v o l v e m e n t i n r e a l i g n m e n t

ACTUAL VS. MINIMUM RESERVES - ITALY

l - 41

~

<—>r-\ E

41 C a

C 0 0 =

41

I

1 5 I I I I I I I I I I I I I I I I I I I I i I I

4 5 6"

t i m e / r e a l i g n m e n t #

P R O B A B I L I T Y OF DEVALUATION D E N M A R K

.a

0.01

]ft 2*-

3 4 5 6

t r m * X r * a i r g n m i * n t **

i t

" indicates involvement in realignment

ACTUAL VS. MINIMUM RESERVES - DENMARK

d

m w

£ vi

gG

= 0

.0

PROBABILITY OF DEVALUATION - FRANCE

_J

m <

m o a:

0.05

0.04-5 -

0.04 -

0.035 -

0.03 -

0.025 -

0.02 -

0.015 -

0.01 -

0.005 -

O

TIME & REALIGNMENT §

* = involvement in realignment

ACTUAL VS. MINIMUM RESERVES - FRANCE

c 0

41 1)

C 41

^ c

= .£

l b

4i

8

170

160 -

150 -

140 -

130 -

120

110 -

100 -

9 0 -

8 0

-&'A^5&' M i l l i o n / F r a n c s

I I I I I 1 2

I I I I I I I I I I I I I I I I I I I I

3 4

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 5 6* TB

t ime/rea lign m e nt #

BEFERENCES

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PROBABILITY OF REALIGNMENTS (page 1)

••; t

. p p r t p

•\ / ^ ::; ,; ::" -y .;

: . ; o } A 'j • ) ;• i ' , -.,-

l>, .1 I.-I t r

i i — , i i V i : i ' 7 . ; -7-U-

'>c; ;' D A O C . . , I •;

• , •' i i

, .. .-•-.;• r

ii'l

- W —

... 0 -j

.... |0 ",

- (•) i

... i.{ j

-

0... A '.'• 8 <096H ""''•) i

0 •'!' 2 1 97ii: "'••' 1 0.-. ^ 3 4 9 7 0 E " 0 1 0 . 3 2 3 6 I 7 E - 0 1 0 , 3 2 03 2 3 F - 0 1

a . . j i 1^

: . '•' • M''fj i i ';

- . ''.:.3'; 0 " 0 '!

-t-rT/4"¹'^ 4ir"0 i P

mil 07 IE:- 01

7301F-01 0.781109 E-

• : •.'••••: (:;i\\:.

l y( j'l) • {-•••" ,

1 67 ;>O0F • 'O'"'

-» ^ ± w .

i -.. i.

7 2 I:

. i J J L X K L i i .

0900 OOF •-') 2

•; ?9C'00E—02

-.? i". <•. ,'.•• : r:

•'>-vt .-•.: f::;.:.

!53145E

.-1 '•'* ; <—•—rf*: - 1 : - ' i ' 'i

i • , ' , • , |r.jj.

}\ i ; ;l. •• \

O i !::

... 0 -j

J .v. i .:.. ».J :

"> £( «!.' ..

"01

.' f ;... -..•' i

•: a:—fV4-

t? t

0, i;^;:::i 5 BOO""'' 1

i ••'t

. ^j .•_. v4 •-.—

PROBABILITY OF REALIGNMENTS (page 2)

.. I:

I H

,',) ;• j ;..' .WW-; |V;j ••

;' : i .-" \ \ \:; !;; I ~C

,21S456E-01 0,93: '••' i

*V i

f. i ,•: .. i ;. ; ..

.J „• L. '..

•v-vF - A

T~:—'—y*T* iv •'": '!",""f." 1 'v' v1':.I "' !"

240001;

2 1 ft i > I > ?•_

"""y,T"j"Ji~'^t'r~*'"7-TT~

207479E~0i

E-01

f !YTtTVTT'T''

• "3000E '{•:.'. !::."'• V.n l'.( "' r?T'.' V >T".'' ? ''',' •

I ) . .*! / '. HIl'.H.-'

• ; ^ ^ A j ; . -r

0,100628 0,.47.81 , 2E-01

If -•-.' • • t-i !-i i-I ! ' t . - y i

i i

r— n. • I 1 •». > >'

A t <i • • *t-

' : • ' > T 6 ' 1 4 d x c . - " t f >\i .—t—Tt*t T - T f t i - i r

vj \v'.',' '.v i"."" '-v 0 0 0 0 E '"• 6

•j 4 VJ'v.-«"«'!.:

: ".'. ; .'' .•-' , ' l"

, , . . , , „ J . y f.\

«LV*

2281 8 BE-Oi 0250'i34''!E~Oi 253382E-01: 0.258264E-01

i i '..; >I *

;28075E-01 .vO'l

" - 0 1

_ ^ _ A ^ - ^:-f-t-;

'•}•; , v { o n : . - o i

* * r , -« "

. •., I

Appendix 1

2 dimensional contour plots of Reserve Limits - R

vs.

shock parameter - b

(used to find initial starting parameters for GQ-OPT)